At time \(t = 0\), a football player kicks a ball from the point \(A\) with position vector ( \(2 \mathbf { i } + \mathbf { j }\) ) m on a horizontal football field. The motion of the ball is modelled as that of a particle moving horizontally with constant velocity \(( 5 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
- the speed of the ball,
- the position vector of the ball after \(t\) seconds.
The point \(B\) on the field has position vector \(( 10 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m }\).
- Find the time when the ball is due north of \(B\).
At time \(t = 0\), another player starts running due north from \(B\) and moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that he intercepts the ball,
- find the value of \(v\).
- State one physical factor, other than air resistance, which would be needed in a refinement of the model of the ball's motion to make the model more realistic.
Turn over
\section*{Paper Reference(s)}
6677/01
\section*{Edexcel GCE }
Examiner's use only
Turn over
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
1.
Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-042_404_755_312_577}
Figure 1 shows the speed-time graph of a cyclist moving on a straight road over a 7 s period. The sections of the graph from \(t = 0\) to \(t = 3\), and from \(t = 3\) to \(t = 7\), are straight lines. The section from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis.
State what can be deduced about the motion of the cyclist from the fact that - the graph from \(t = 0\) to \(t = 3\) is a straight line,
- the graph from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis.
- Find the distance travelled by the cyclist during this 7 s period.
2. Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. They are moving in opposite directions on a smooth horizontal table and collide directly. Immediately before the collision, the speed of \(A\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As a result of the collision, the direction of motion of \(B\) is reversed and its speed immediately after the collision is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find - the speed of \(A\) immediately after the collision, stating clearly whether the direction of motion of \(A\) is changed by the collision,
- the magnitude of the impulse exerted on \(B\) in the collision, stating clearly the units in which your answer is given.
3. A train moves along a straight track with constant acceleration. Three telegraph poles are set at equal intervals beside the track at points \(A , B\) and \(C\), where \(A B = 50 \mathrm {~m}\) and \(B C = 50 \mathrm {~m}\). The front of the train passes \(A\) with speed \(22.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it passes \(B\). Find - the acceleration of the train,
- the speed of the front of the train when it passes \(C\),
- the time that elapses from the instant the front of the train passes \(B\) to the instant it passes \(C\).
4.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-045_273_611_319_676}
\end{figure}
A particle \(P\) of mass 0.5 kg is on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The particle is held at rest on the plane by the action of a force of magnitude 4 N acting up the plane in a direction parallel to a line of greatest slope of the plane, as shown in Figure 2. The particle is on the point of slipping up the plane. - Find the coefficient of friction between \(P\) and the plane.
The force of magnitude 4 N is removed.
- Find the acceleration of \(P\) down the plane.
5.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-047_237_805_303_598}
\end{figure}
A steel girder \(A B\) has weight 210 N . It is held in equilibrium in a horizontal position by two vertical cables. One cable is attached to the end \(A\). The other cable is attached to the point \(C\) on the girder, where \(A C = 90 \mathrm {~cm}\), as shown in Figure 3. The girder is modelled as a uniform rod, and the cables as light inextensible strings.
Given that the tension in the cable at \(C\) is twice the tension in the cable at \(A\), find - the tension in the cable at \(A\),
- show that \(A B = 120 \mathrm {~cm}\).
A small load of weight \(W\) newtons is attached to the girder at \(B\). The load is modelled as a particle. The girder remains in equilibrium in a horizontal position. The tension in the cable at \(C\) is now three times the tension in the cable at \(A\).
- Find the value of \(W\).
- A car is towing a trailer along a straight horizontal road by means of a horizontal tow-rope. The mass of the car is 1400 kg . The mass of the trailer is 700 kg . The car and the trailer are modelled as particles and the tow-rope as a light inextensible string. The resistances to motion of the car and the trailer are assumed to be constant and of magnitude 630 N and 280 N respectively. The driving force on the car, due to its engine, is 2380 N . Find
- the acceleration of the car,
- the tension in the tow-rope.
When the car and trailer are moving at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the tow-rope breaks. Assuming that the driving force on the car and the resistances to motion are unchanged, - find the distance moved by the car in the first 4 s after the tow-rope breaks.
(6) - State how you have used the modelling assumption that the tow-rope is inextensible.
- \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and north respectively.]
A ship \(S\) is moving with constant velocity \(( - 2.5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time 1200, the position vector of \(S\) relative to a fixed origin \(O\) is \(( 16 \mathbf { i } + 5 \mathbf { j } )\) km. Find - the speed of \(S\),
- the bearing on which \(S\) is moving.
The ship is heading directly towards a submerged rock \(R\). A radar tracking station calculates that, if \(S\) continues on the same course with the same speed, it will hit \(R\) at the time 1500.
- Find the position vector of \(R\).
The tracking station warns the ship's captain of the situation. The captain maintains \(S\) on its course with the same speed until the time is 1400 . He then changes course so that \(S\) moves due north at a constant speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Assuming that \(S\) continues to move with this new constant velocity, find
- an expression for the position vector of the ship \(t\) hours after 1400,
- the time when \(S\) will be due east of \(R\),
- the distance of \(S\) from \(R\) at the time 1600.
\end{table}
Turn over
1.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-054_287_625_310_662}
\end{figure}
A particle of weight 24 N is held in equilibrium by two light inextensible strings. One string is horizontal. The other string is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 1. The tension in the horizontal string is \(Q\) newtons and the tension in the other string is \(P\) newtons. Find - the value of \(P\),
- the value of \(Q\).
2.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-055_246_652_310_653}
\end{figure}
A uniform plank \(A B\) has weight 120 N and length 3 m . The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(C D = x \mathrm {~m}\), as shown in Figure 2. The reaction of the support on the plank at \(D\) has magnitude 80 N . Modelling the plank as a rod, - show that \(x = 0.75\)
A rock is now placed at \(B\) and the plank is on the point of tilting about \(D\). Modelling the rock as a particle, find
- the weight of the rock,
- the magnitude of the reaction of the support on the plank at \(D\).
- State how you have used the model of the rock as a particle.
- A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. When \(t = 0 , P\) has velocity ( \(3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and at time \(t = 4 \mathrm {~s} , P\) has velocity \(( 15 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
- the acceleration of \(P\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
- the magnitude of \(\mathbf { F }\),
- the velocity of \(P\) at time \(t = 6 \mathrm {~s}\).
- A particle \(P\) of mass 0.3 kg is moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal table. The particle \(P\) collides directly with a particle \(Q\) of mass 0.6 kg , which is at rest on the table. Immediately after the particles collide, \(P\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is reversed by the collision. Find
- the value of \(u\),
- the magnitude of the impulse exerted by \(P\) on \(Q\).
Immediately after the collision, a constant force of magnitude \(R\) newtons is applied to \(Q\) in the direction directly opposite to the direction of motion of \(Q\). As a result \(Q\) is brought to rest in 1.5 s . - Find the value of \(R\).
- A ball is projected vertically upwards with speed \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\), which is 1.5 m above the ground. After projection, the ball moves freely under gravity until it reaches the ground. Modelling the ball as a particle, find
- the greatest height above \(A\) reached by the ball,
- the speed of the ball as it reaches the ground,
- the time between the instant when the ball is projected from \(A\) and the instant when the ball reaches the ground.
6.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-062_230_642_298_659}
\end{figure}
A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope. The rope makes an angle of \(20 ^ { \circ }\) with the ground, as shown in Figure 3. The coefficient of friction between the box and the ground is 0.4 . The box is modelled as a particle and the rope as a light, inextensible string. The tension in the rope is \(P\) newtons. - Find the value of \(P\).
The tension in the rope is now increased to 150 N .
- Find the acceleration of the box.
7.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-064_465_1182_301_420}
\end{figure}
Figure 4 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a small smooth light pulley \(A\) fixed at the top of the plane. The part of the string from \(P\) to \(A\) is parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below \(A\). The system is released from rest with the string taut. - Write down an equation of motion for \(P\) and an equation of motion for \(Q\).
- Hence show that the acceleration of \(Q\) is \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
- Find the tension in the string.
- State where in your calculations you have used the information that the string is inextensible.
On release, \(Q\) is at a height of 0.8 m above the ground. When \(Q\) reaches the ground, it is brought to rest immediately by the impact with the ground and does not rebound. The initial distance of \(P\) from \(A\) is such that in the subsequent motion \(P\) does not reach \(A\). Find
- the speed of \(Q\) as it reaches the ground,
- the time between the instant when \(Q\) reaches the ground and the instant when the string becomes taut again.
Turn over
1.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-067_579_490_301_730}
\end{figure}
A particle \(P\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 12 N is applied to \(P\). The particle \(P\) is in equilibrium with the string taut and \(O P\) making an angle of \(20 ^ { \circ }\) with the downward vertical, as shown in Figure 1.
Find - the tension in the string,
- the weight of \(P\).
2. Two particles \(A\) and \(B\), of mass 0.3 kg and \(m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line so that the particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. In the collision the direction of motion of each particle is reversed and, immediately after the collision, the speed of each particle is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find - the magnitude of the impulse exerted by \(B\) on \(A\) in the collision,
- the value of \(m\).
3.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-069_282_842_296_561}
\end{figure}
A uniform rod \(A B\) has length 1.5 m and mass 8 kg . A particle of mass \(m \mathrm {~kg}\) is attached to the rod at \(B\). The rod is supported at the point \(C\), where \(A C = 0.9 \mathrm {~m}\), and the system is in equilibrium with \(A B\) horizontal, as shown in Figure 2. - Show that \(m = 2\).
A particle of mass 5 kg is now attached to the rod at \(A\) and the support is moved from \(C\) to a point \(D\) of the rod. The system, including both particles, is again in equilibrium with \(A B\) horizontal.
- Find the distance \(A D\).
- A car is moving along a straight horizontal road. At time \(t = 0\), the car passes a point \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car moves with constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until \(t = 10 \mathrm {~s}\). The car then decelerates uniformly for 8 s . At time \(t = 18 \mathrm {~s}\), the speed of the car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and this speed is maintained until the car reaches the point \(B\) at time \(t = 30 \mathrm {~s}\).
- Sketch, in the space below, a speed-time graph to show the motion of the car from \(A\) to \(B\).
Given that \(A B = 526 \mathrm {~m}\), find - the value of \(V\),
- the deceleration of the car between \(t = 10 \mathrm {~s}\) and \(t = 18 \mathrm {~s}\).
5.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-073_218_479_287_744}
\end{figure}
A small ring of mass 0.25 kg is threaded on a fixed rough horizontal rod. The ring is pulled upwards by a light string which makes an angle \(40 ^ { \circ }\) with the horizontal, as shown in Figure 3. The string and the rod are in the same vertical plane. The tension in the string is 1.2 N and the coefficient of friction between the ring and the rod is \(\mu\). Given that the ring is in limiting equilibrium, find - the normal reaction between the ring and the rod,
- the value of \(\mu\).
6.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-075_572_586_299_696}
\end{figure}
Two particles \(P\) and \(Q\) have mass 0.5 kg and \(m \mathrm {~kg}\) respectively, where \(m < 0.5\). The particles are connected by a light inextensible string which passes over a smooth, fixed pulley. Initially \(P\) is 3.15 m above horizontal ground. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 4. After \(P\) has been descending for 1.5 s , it strikes the ground. Particle \(P\) reaches the ground before \(Q\) has reached the pulley. - Show that the acceleration of \(P\) as it descends is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
- Find the tension in the string as \(P\) descends.
- Show that \(m = \frac { 5 } { 18 }\).
- State how you have used the information that the string is inextensible.
When \(P\) strikes the ground, \(P\) does not rebound and the string becomes slack. Particle \(Q\) then moves freely under gravity, without reaching the pulley, until the string becomes taut again.
- Find the time between the instant when \(P\) strikes the ground and the instant when the string becomes taut again.
- A boat \(B\) is moving with constant velocity. At noon, \(B\) is at the point with position vector \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { km }\) with respect to a fixed origin \(O\). At 1430 on the same day, \(B\) is at the point with position vector \(( 8 \mathbf { i } + 11 \mathbf { j } ) \mathrm { km }\).
- Find the velocity of \(B\), giving your answer in the form \(p \mathbf { i } + q \mathbf { j }\).
At time \(t\) hours after noon, the position vector of \(B\) is \(\mathbf { b } \mathrm { km }\). - Find, in terms of \(t\), an expression for \(\mathbf { b }\).
Another boat \(C\) is also moving with constant velocity. The position vector of \(C\), \(\mathbf { c k m }\), at time \(t\) hours after noon, is given by
$$\mathbf { c } = ( - 9 \mathbf { i } + 20 \mathbf { j } ) + t ( 6 \mathbf { i } + \lambda \mathbf { j } ) ,$$
where \(\lambda\) is a constant. Given that \(C\) intercepts \(B\),
- find the value of \(\lambda\),
- show that, before \(C\) intercepts \(B\), the boats are moving with the same speed.
Turn over
advancing learning, changing lives
- Two particles \(A\) and \(B\) have masses 4 kg and \(m \mathrm {~kg}\) respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(A\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Find the magnitude of the impulse exerted on \(A\) in the collision.
Immediately after the collision, the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). - Find the value of \(m\).
2. A firework rocket starts from rest at ground level and moves vertically. In the first 3 s of its motion, the rocket rises 27 m . The rocket is modelled as a particle moving with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find - the value of \(a\),
- the speed of the rocket 3 s after it has left the ground.
After 3 s , the rocket burns out. The motion of the rocket is now modelled as that of a particle moving freely under gravity.
- Find the height of the rocket above the ground 5 s after it has left the ground.
3. A car moves along a horizontal straight road, passing two points \(A\) and \(B\). At \(A\) the speed of the car is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the driver passes \(A\), he sees a warning sign \(W\) ahead of him, 120 m away. He immediately applies the brakes and the car decelerates with uniform deceleration, reaching \(W\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At \(W\), the driver sees that the road is clear. He then immediately accelerates the car with uniform acceleration for 16 s to reach a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 } ( V > 15 )\). He then maintains the car at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Moving at this constant speed, the car passes \(B\) after a further 22 s . - Sketch, in the space below, a speed-time graph to illustrate the motion of the car as it moves from \(A\) to \(B\).
- Find the time taken for the car to move from \(A\) to \(B\).
The distance from \(A\) to \(B\) is 1 km .
- Find the value of \(V\).
4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-084_305_607_246_701}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
A particle \(P\) of mass 6 kg lies on the surface of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 49 N , acting at an angle \(\theta\) to the plane, as shown in Figure 1. The force acts in a vertical plane through a line of greatest slope of the plane. - Show that \(\cos \theta = \frac { 3 } { 5 }\).
- Find the normal reaction between \(P\) and the plane.
The direction of the force of magnitude 49 N is now changed. It is now applied horizontally to \(P\) so that \(P\) moves up the plane. The force again acts in a vertical plane through a line of greatest slope of the plane.
- Find the initial acceleration of \(P\).
\(\_\_\_\_\)}
5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-086_315_817_255_587}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A beam \(A B\) has mass 12 kg and length 5 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\), the other to the point \(C\) on the beam, where \(B C = 1 \mathrm {~m}\), as shown in Figure 2. The beam is modelled as a uniform rod, and the ropes as light strings. - Find
- the tension in the rope at \(C\),
- the tension in the rope at \(A\).
A small load of mass 16 kg is attached to the beam at a point which is \(y\) metres from \(A\). The load is modelled as a particle. Given that the beam remains in equilibrium in a horizontal position,
- find, in terms of \(y\), an expression for the tension in the rope at \(C\).
The rope at \(C\) will break if its tension exceeds 98 N. The rope at \(A\) cannot break.
- Find the range of possible positions on the beam where the load can be attached without the rope at \(C\) breaking.
6. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]
A particle \(P\) is moving with constant velocity \(( - 5 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
- the speed of \(P\),
- the direction of motion of \(P\), giving your answer as a bearing.
At time \(t = 0 , P\) is at the point \(A\) with position vector ( \(7 \mathbf { i } - 10 \mathbf { j }\) ) m relative to a fixed origin \(O\). When \(t = 3 \mathrm {~s}\), the velocity of \(P\) changes and it moves with velocity \(( u \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(u\) and \(v\) are constants. After a further 4 s , it passes through \(O\) and continues to move with velocity ( \(u \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
- Find the values of \(u\) and \(v\).
- Find the total time taken for \(P\) to move from \(A\) to a position which is due south of A.
7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-090_292_897_278_415}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough horizontal table. The string passes over a small smooth pulley \(P\) fixed on the edge of the table. The particle \(B\) hangs freely below the pulley, as shown in Figure 3. The coefficient of friction between \(A\) and the table is \(\mu\). The particles are released from rest with the string taut. Immediately after release, the magnitude of the acceleration of \(A\) and \(B\) is \(\frac { 4 } { 9 } g\). By writing down separate equations of motion for \(A\) and \(B\), - find the tension in the string immediately after the particles begin to move,
- show that \(\mu = \frac { 2 } { 3 }\).
When \(B\) has fallen a distance \(h\), it hits the ground and does not rebound. Particle \(A\) is then a distance \(\frac { 1 } { 3 } h\) from \(P\).
- Find the speed of \(A\) as it reaches \(P\).
- State how you have used the information that the string is light.
Turn over
advancing learning, changing lives
- Two particles \(P\) and \(Q\) have mass 0.4 kg and 0.6 kg respectively. The particles are initially at rest on a smooth horizontal table. Particle \(P\) is given an impulse of magnitude 3 N s in the direction \(P Q\).
- Find the speed of \(P\) immediately before it collides with \(Q\).
Immediately after the collision between \(P\) and \(Q\), the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). - Show that immediately after the collision \(P\) is at rest.
2. At time \(t = 0\), a particle is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 10 m above the ground. At time \(T\) seconds, the particle hits the ground with speed \(17.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find - the value of \(u\),
- the value of \(T\).
3. A particle \(P\) of mass 0.4 kg moves under the action of a single constant force \(\mathbf { F }\) newtons. The acceleration of \(P\) is \(( 6 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find - the angle between the acceleration and \(\mathbf { i }\),
- the magnitude of \(\mathbf { F }\).
At time \(t\) seconds the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Given that when \(t = 0 , \mathbf { v } = 9 \mathbf { i } - 10 \mathbf { j }\), (c) find the velocity of \(P\) when \(t = 5\).
4. A car is moving along a straight horizontal road. The speed of the car as it passes the point \(A\) is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car maintains this speed for 30 s . The car then decelerates uniformly to a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is then maintained until the car passes the point \(B\). The time taken to travel from \(A\) to \(B\) is 90 s and \(A B = 1410 \mathrm {~m}\). - Sketch, in the space below, a speed-time graph to show the motion of the car from \(A\) to \(B\).
- Calculate the deceleration of the car as it decelerates from \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Question 4 continued \(\_\_\_\_\)
5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-098_357_968_274_484}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Two forces \(\mathbf { P }\) and \(\mathbf { Q }\) act on a particle at a point \(O\). The force \(\mathbf { P }\) has magnitude 15 N and the force \(\mathbf { Q }\) has magnitude \(X\) newtons. The angle between \(\mathbf { P }\) and \(\mathbf { Q }\) is \(150 ^ { \circ }\), as shown in Figure 1. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(\mathbf { R }\).
Given that the angle between \(\mathbf { R }\) and \(\mathbf { Q }\) is \(50 ^ { \circ }\), find - the magnitude of \(\mathbf { R }\),
- the value of \(X\).
6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-099_392_678_260_614}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A plank \(A B\) has mass 12 kg and length 2.4 m . A load of mass 8 kg is attached to the plank at the point \(C\), where \(A C = 0.8 \mathrm {~m}\). The loaded plank is held in equilibrium, with \(A B\) horizontal, by two vertical ropes, one attached at \(A\) and the other attached at \(B\), as shown in Figure 2. The plank is modelled as a uniform rod, the load as a particle and the ropes as light inextensible strings. - Find the tension in the rope attached at \(B\).
The plank is now modelled as a non-uniform rod. With the new model, the tension in the rope attached at \(A\) is 10 N greater than the tension in the rope attached at \(B\).
- Find the distance of the centre of mass of the plank from \(A\).
7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-101_291_726_265_607}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
A package of mass 4 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The package is held in equilibrium by a force of magnitude 45 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 3. The force is acting in a vertical plane through a line of greatest slope of the plane. The package is in equilibrium on the point of moving up the plane. The package is modelled as a particle. Find - the magnitude of the normal reaction of the plane on the package,
- the coefficient of friction between the plane and the package.
8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-103_131_940_269_498}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force \(\mathbf { F }\) of magnitude 30 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 4. The force is applied for 3 s and during this time \(Q\) travels a distance of 6 m . The coefficient of friction between each particle and the plane is \(\mu\). Find - the acceleration of \(Q\),
- the value of \(\mu\),
- the tension in the string.
- State how in your calculation you have used the information that the string is inextensible.
When the particles have moved for 3 s , the force \(\mathbf { F }\) is removed.
- Find the time between the instant that the force is removed and the instant that \(Q\) comes to rest.
Paper Reference(s)
\section*{6677/01}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Examiner's use only}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-117_97_310_495_1635}
\end{figure}
\(\mathbf { F } _ { 1 } = ( \mathbf { i } - 3 \mathbf { j } ) \mathrm { N }\),
\(\mathbf { F } _ { 2 } = ( p \mathbf { i } + 2 p \mathbf { j } ) \mathrm { N }\), where \(p\) is a positive constant. - Find the angle between \(\mathbf { F } _ { 2 }\) and \(\mathbf { j }\).
\end{enumerate}
The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(\mathbf { R }\). Given that \(\mathbf { R }\) is parallel to \(\mathbf { i }\),
- find the value of \(p\).
3. Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(2 m\) and the mass of \(B\) is \(m\). The particles are moving along the same straight line but in opposite directions and they collide directly. Immediately before they collide the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(3 u\). The magnitude of the impulse received by each particle in the collision is \(\frac { 7 m u } { 2 }\).
Find - the speed of \(A\) immediately after the collision,
- the speed of \(B\) immediately after the collision.
4. A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 4 } { 3 }\), and released from rest. The coefficient of friction between the brick and the plane is \(\frac { 1 } { 3 }\).
Find the acceleration of the brick.
(9)
5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-122_332_780_292_585}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
A small box of mass 15 kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2 . Aforce of magnitude \(P\) newtons is applied to the box at \(50 ^ { \circ }\) to the horizontal, as shown in Figure 1. The box is on the point of sliding along the plane.
Find the value of \(P\), giving your answer to 2 significant figures.
6. A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N . Find - the acceleration of the car and trailer,
- the magnitude of the tension in the towbar.
The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude \(F\) newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N ,
- find the value of \(F\).
7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-125_337_1287_228_370}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A beam \(A B\) is supported by two vertical ropes, which are attached to the beam at points \(P\) and \(Q\), where \(A P = 0.3 \mathrm {~m}\) and \(B Q = 0.3 \mathrm {~m}\). The beam is modelled as a uniform rod, of length 2 m and mass 20 kg . The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between \(P\) and \(Q\). The gymnast is modelled as a particle attached to the beam at the point \(X\), where \(P X = x \mathrm {~m} , 0 < x < 1.4\) as shown in Figure 2. The beam rests in equilibrium in a horizontal position. - Show that the tension in the rope attached to the beam at \(P\) is \(( 588 - 350 x ) \mathrm { N }\).
- Find, in terms of \(x\), the tension in the rope attached to the beam at \(Q\).
- Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope.
Given that the tension in the rope attached at \(Q\) is three times the tension in the rope attached at \(P\),
- find the value of \(x\).
\section*{LU
\(\_\_\_\_\)}
- \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.] A hiker \(H\) is walking with constant velocity \(( 1.2 \mathbf { i } - 0.9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
- Find the speed of \(H\).
(2)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-127_599_1057_521_445}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
A horizontal field \(O A B C\) is rectangular with \(O A\) due east and \(O C\) due north, as shown in Figure 3. At twelve noon hiker \(H\) is at the point \(Y\) with position vector \(100 \mathbf { j } \mathrm {~m}\), relative to the fixed origin \(O\). - Write down the position vector of \(H\) at time \(t\) seconds after noon.
At noon, another hiker \(K\) is at the point with position vector \(( 9 \mathbf { i } + 46 \mathbf { j } )\) m. Hiker \(K\) is moving with constant velocity \(( 0.75 \mathbf { i } + 1.8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
- Show that, at time \(t\) seconds after noon,
$$\overrightarrow { H K } = [ ( 9 - 0.45 t ) \mathbf { i } + ( 2.7 t - 54 ) \mathbf { j } ] \text { metres. }$$
Hence,
- show that the two hikers meet and find the position vector of the point where they meet.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{physicsandmathstutor.com}
\end{table}
Paper Reference(s)
6677/01
\section*{Edexcel GCE }
Examiner's use only
Turn over
advancing learning, changing lives
- A particle \(A\) of mass 2 kg is moving along a straight horizontal line with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(B\) of mass \(m \mathrm {~kg}\) is moving along the same straight line, in the opposite direction to \(A\), with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particles collide. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, \(A\) is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
- the magnitude of the impulse exerted by \(B\) on \(A\) in the collision,
- the value of \(m\).
- An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This speed is then maintained for \(T\) seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s .
- In the space below, sketch a speed-time graph to illustrate the motion of the athlete.
- Calculate the value of \(T\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-132_271_750_214_598}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
A particle of mass \(m \mathrm {~kg}\) is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively, as shown in Figure 1.
Given that the tension in \(A C\) is 20 N , find - the tension in \(B C\),
- the value of \(m\).
4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-133_557_673_127_646}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A pole \(A B\) has length 3 m and weight \(W\) newtons. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points \(A\) and \(C\) where \(A C = 1.8 \mathrm {~m}\), as shown in Figure 2. A load of weight 20 N is attached to the rod at \(B\). The pole is modelled as a uniform rod, the ropes as light inextensible strings and the load as a particle. - Show that the tension in the rope attached to the pole at \(C\) is \(\left( \frac { 5 } { 6 } W + \frac { 100 } { 3 } \right) \mathrm { N }\).
- Find, in terms of \(W\), the tension in the rope attached to the pole at \(A\).
Given that the tension in the rope attached to the pole at \(C\) is eight times the tension in the rope attached to the pole at \(A\),
- find the value of \(W\).
- A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find
- the acceleration of the particle,
- the coefficient of friction between the particle and the plane.
The particle is now held on the same rough plane by a horizontal force of magnitude \(X\) newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-135_255_725_890_621}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure} - Find the value of \(X\).
6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-137_519_537_210_708}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Two particles \(A\) and \(B\) have masses \(5 m\) and \(k m\) respectively, where \(k < 5\). The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut, the hanging parts of the string vertical and with \(A\) and \(B\) at the same height above a horizontal plane, as shown in Figure 4. The system is released from rest. After release, \(A\) descends with acceleration \(\frac { 1 } { 4 } g\). - Show that the tension in the string as \(A\) descends is \(\frac { 15 } { 4 } \mathrm { mg }\).
- Find the value of \(k\).
- State how you have used the information that the pulley is smooth.
After descending for 1.2 s , the particle \(A\) reaches the plane. It is immediately brought to rest by the impact with the plane. The initial distance between \(B\) and the pulley is such that, in the subsequent motion, \(B\) does not reach the pulley.
- Find the greatest height reached by \(B\) above the plane.
7. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.]
A ship \(S\) is moving along a straight line with constant velocity. At time \(t\) hours the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\). When \(t = 0 , \mathbf { s } = 9 \mathbf { i } - 6 \mathbf { j }\). When \(t = 4 , \mathbf { s } = 21 \mathbf { i } + 10 \mathbf { j }\). Find - the speed of \(S\),
- the direction in which \(S\) is moving, giving your answer as a bearing.
- Show that \(\mathbf { s } = ( 3 t + 9 ) \mathbf { i } + ( 4 t - 6 ) \mathbf { j }\).
A lighthouse \(L\) is located at the point with position vector \(( 18 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km }\). When \(t = T\), the ship \(S\) is 10 km from \(L\).
- Find the possible values of \(T\).
Turn over
advancing learning, changing lives
- A particle \(P\) is moving with constant velocity \(( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 6 \mathrm {~s} P\) is at the point with position vector \(( - 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }\). Find the distance of \(P\) from the origin at time \(t = 2 \mathrm {~s}\).
(5)
- Particle \(P\) has mass \(m \mathrm {~kg}\) and particle \(Q\) has mass \(3 m \mathrm {~kg}\). The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(k u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved.
- Find the value of \(k\).
- Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-144_195_579_260_507}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is \(\frac { 1 } { 2 }\). The box is pushed by a force of magnitude 100 N which acts at an angle of \(30 ^ { \circ }\) with the floor, as shown in Figure 1.
Given that the box moves with constant speed, find the mass of the box.
4. A beam \(A B\) has length 6 m and weight 200 N . The beam rests in a horizontal position on two supports at the points \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\). Two children, Sophie and Tom, each of weight 500 N , stand on the beam with Sophie standing twice as far from the end \(B\) as Tom. The beam remains horizontal and in equilibrium and the magnitude of the reaction at \(D\) is three times the magnitude of the reaction at \(C\). By modelling the beam as a uniform rod and the two children as particles, find how far Tom is standing from the end \(B\).
5. Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t = 0 , P\) overtakes \(Q\) which is moving with constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). From \(t = T\) seconds, P decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25 \mathrm {~s}\), \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\). - Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\).
- Find the value of \(T\).
6. A ball is projected vertically upwards with a speed of \(14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find - the greatest height, above the ground, reached by the ball,
- the speed with which the ball first strikes the ground,
- the total time from when the ball is projected to when it first strikes the ground.
7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-150_275_712_269_612}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude \(P\) newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle \(\alpha\), where tan \(\alpha = \frac { 3 } { 4 }\), as shown in Figure 2.
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 3 }\).
Given that the particle is on the point of sliding up the plane, find - the magnitude of the normal reaction between the particle and the plane,
- the value of \(P\).
8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-152_890_428_237_754}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion \(B\) does not reach the pulley. - Find the tension in the string immediately after the particles are released.
- Find the acceleration of \(A\) immediately after the particles are released.
When the particles have been moving for 0.5 s , the string breaks.
- Find the further time that elapses until \(B\) hits the floor.
Turn over
advancing learning, changing lives
- Two particles \(B\) and \(C\) have mass \(m \mathrm {~kg}\) and 3 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table. The two particles collide directly. Immediately before the collision, the speed of \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the collision the direction of motion of \(C\) is reversed and the direction of motion of \(B\) is unchanged. Immediately after the collision, the speed of \(B\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(C\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find - the value of \(m\),
- the magnitude of the impulse received by \(C\).
2. A ball is thrown vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(P\) at height \(h\) metres above the ground. The ball hits the ground 0.75 s later. The speed of the ball immediately before it hits the ground is \(6.45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is modelled as a particle. - Show that \(u = 0.9\)
- Find the height above \(P\) to which the ball rises before it starts to fall towards the ground again.
- Find the value of \(h\).
3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-157_245_860_260_543}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
A uniform beam \(A B\) has mass 20 kg and length 6 m . The beam rests in equilibrium in a horizontal position on two smooth supports. One support is at \(C\), where \(A C = 1 \mathrm {~m}\), and the other is at the end \(B\), as shown in Figure 1. The beam is modelled as a rod. - Find the magnitudes of the reactions on the beam at \(B\) and at \(C\).
A boy of mass 30 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The magnitudes of the reactions on the beam at \(B\) and at \(C\) are now equal. The boy is modelled as a particle.
- Find the distance \(A D\).
- A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. The velocity of \(P\) is \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 0\), and \(( 7 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 5 \mathrm {~s}\).
Find - the speed of \(P\) at \(t = 0\),
- the vector \(\mathbf { F }\) in the form \(a \mathbf { i } + b \mathbf { j }\),
- the value of \(t\) when \(P\) is moving parallel to \(\mathbf { i }\).
- A car accelerates uniformly from rest for 20 seconds. It moves at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 40 seconds and then decelerates uniformly for 10 seconds until it comes to rest.
- For the motion of the car, sketch
- a speed-time graph,
- an acceleration-time graph.
Given that the total distance moved by the car is 880 m , - find the value of \(v\).
6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-163_426_768_239_653}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A particle of weight 120 N is placed on a fixed rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\).
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
The particle is held at rest in equilibrium by a horizontal force of magnitude 30 N , which acts in the vertical plane containing the line of greatest slope of the plane through the particle, as shown in Figure 2. - Show that the normal reaction between the particle and the plane has magnitude 114 N .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-163_433_774_1464_604}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The horizontal force is removed and replaced by a force of magnitude \(P\) newtons acting up the slope along the line of greatest slope of the plane through the particle, as shown in Figure 3. The particle remains in equilibrium. - Find the greatest possible value of \(P\).
- Find the magnitude and direction of the frictional force acting on the particle when \(P = 30\).
7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-165_590_1217_226_367}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Two particles \(A\) and \(B\), of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially \(B\) is held at rest on a rough fixed plane inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The part of the string from \(B\) to \(P\) is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, \(P\), fixed at the top of the plane. The particle \(A\) hangs freely below \(P\), as shown in Figure 4. The coefficient of friction between \(B\) and the plane is \(\frac { 2 } { 3 }\). The particles are released from rest with the string taut and \(B\) moves up the plane. - Find the magnitude of the acceleration of \(B\) immediately after release.
- Find the speed of \(B\) when it has moved 1 m up the plane.
When \(B\) has moved 1 m up the plane the string breaks. Given that in the subsequent motion \(B\) does not reach \(P\),
- find the time between the instants when the string breaks and when \(B\) comes to instantaneous rest.
Turn over
advancing learning, changing lives
- At time \(t = 0\) a ball is projected vertically upwards from a point \(O\) and rises to a maximum height of 40 m above \(O\). The ball is modelled as a particle moving freely under gravity.
- Show that the speed of projection is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Find the times, in seconds, when the ball is 33.6 m above \(O\).
- Particle \(P\) has mass 3 kg and particle \(Q\) has mass 2 kg . The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, \(P\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, both particles move in the same direction and the difference in their speeds is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Find the speed of each particle after the collision.
- Find the magnitude of the impulse exerted on \(P\) by \(Q\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-170_344_771_221_589}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
A particle of weight \(W\) newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N . The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 1.
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
Given that the particle is on the point of sliding down the plane,
(i) show that the magnitude of the normal reaction between the particle and the plane is 20 N ,
(ii) find the value of \(W\).
- A girl runs a 400 m race in a time of 84 s . In a model of this race, it is assumed that, starting from rest, she moves with constant acceleration for 4 s , reaching a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She maintains this speed for 60 s and then moves with constant deceleration for 20 s , crossing the finishing line with a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race.
- Find the distance run by the girl in the first 64 s of the race.
- Find the value of \(V\).
- Find the deceleration of the girl in the final 20 s of her race.
- A plank \(P Q R\), of length 8 m and mass 20 kg , is in equilibrium in a horizontal position on two supports at \(P\) and \(Q\), where \(P Q = 6 \mathrm {~m}\).
A child of mass 40 kg stands on the plank at a distance of 2 m from \(P\) and a block of mass \(M \mathrm {~kg}\) is placed on the plank at the end \(R\). The plank remains horizontal and in equilibrium. The force exerted on the plank by the support at \(P\) is equal to the force exerted on the plank by the support at \(Q\).
By modelling the plank as a uniform rod, and the child and the block as particles, - find the magnitude of the force exerted on the plank by the support at \(P\),
- find the value of \(M\).
- State how, in your calculations, you have used the fact that the child and the block can be modelled as particles.
6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-176_369_954_214_497}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Two particles \(P\) and \(Q\) have masses 0.3 kg and \(m \mathrm {~kg}\) respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\).
The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle \(P\) is held at rest on the inclined plane and the particle \(Q\) hangs freely below the pulley with the string taut, as shown in Figure 2.
The system is released from rest and \(Q\) accelerates vertically downwards at \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find - the magnitude of the normal reaction of the inclined plane on \(P\),
- the value of \(m\).
When the particles have been moving for 0.5 s , the string breaks. Assuming that \(P\) does not reach the pulley,
- find the further time that elapses until \(P\) comes to instantaneous rest.
- \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin \(O\).]
Two ships \(P\) and \(Q\) are moving with constant velocities. Ship \(P\) moves with velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and ship \(Q\) moves with velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). - Find, to the nearest degree, the bearing on which \(Q\) is moving.
At 2 pm , ship \(P\) is at the point with position vector \(( \mathbf { i } + \mathbf { j } ) \mathrm { km }\) and \(\operatorname { ship } Q\) is at the point with position vector \(( - 2 \mathbf { j } ) \mathrm { km }\).
At time \(t\) hours after 2 pm , the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\) and the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\).
- Write down expressions, in terms of \(t\), for
- \(\mathbf { p }\),
- \(\mathbf { q }\),
- \(\overrightarrow { P Q }\).
- Find the time when
- \(Q\) is due north of \(P\),
- \(Q\) is north-west of \(P\).