- A particle \(P\) of mass 1.5 kg is moving along a straight horizontal line with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(Q\) of mass 2.5 kg is moving, in the opposite direction, along the same straight line with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particles collide. Immediately after the collision the direction of motion of \(P\) is reversed and its speed is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Calculate the speed of \(Q\) immediately after the impact.
- State whether or not the direction of motion of \(Q\) is changed by the collision.
- Calculate the magnitude of the impulse exerted by \(Q\) on \(P\), giving the units of your answer.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-004_521_858_306_552}
\end{figure}
A plank \(A B\) has mass 40 kg and length 3 m . A load of mass 20 kg is attached to the plank at \(B\). The loaded plank is held in equilibrium, with \(A B\) horizontal, by two vertical ropes attached at \(A\) and \(C\), as shown in Figure 1. The plank is modelled as a uniform rod and the load as a particle. Given that the tension in the rope at \(C\) is three times the tension in the rope at \(A\), calculate - the tension in the rope at \(C\),
- the distance \(C B\).
3.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-005_851_1073_312_456}
\end{figure}
A sprinter runs a race of 200 m . Her total time for running the race is 25 s . Figure 2 is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 s . The speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is maintained for 16 s and she then decelerates uniformly to a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of the race. Calculate - the distance covered by the sprinter in the first 20 s of the race,
- the value of \(u\),
- the deceleration of the sprinter in the last 5 s of the race.
4.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-007_330_675_287_644}
\end{figure}
A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate - the normal reaction of the plane on \(P\),
- the value of \(X\).
The force of magnitude \(X\) newtons is now removed.
- Show that \(P\) remains in equilibrium on the plane.
5.
Figure 4
\includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-009_609_1026_301_516}
A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s . Modelling \(A\) and \(B\) as particles, calculate - the acceleration of \(B\),
- the tension in the string,
- the value of \(\mu\).
- State how in your calculations you have used the information that the string is inextensible.
- A stone \(S\) is sliding on ice. The stone is moving along a straight horizontal line \(A B C\), where \(A B = 24 \mathrm {~m}\) and \(A C = 30 \mathrm {~m}\). The stone is subject to a constant resistance to motion of magnitude 0.3 N . At \(A\) the speed of \(S\) is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at \(B\) the speed of \(S\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate
- the deceleration of \(S\),
- the speed of \(S\) at \(C\).
- Show that the mass of \(S\) is 0.1 kg .
At \(C\), the stone \(S\) hits a vertical wall, rebounds from the wall and then slides back along the line \(C A\). The magnitude of the impulse of the wall on \(S\) is 2.4 Ns and the stone continues to move against a constant resistance of 0.3 N . - Calculate the time between the instant that \(S\) rebounds from the wall and the instant that \(S\) comes to rest.
7. Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \(( 14 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km }\). Three hours later, \(P\) is at the point with position vector \(( 29 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). The ship \(Q\) travels with velocity \(12 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively. Find - the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
- expressions for \(\mathbf { p }\) and \(\mathbf { q }\), in terms of \(t\), i and \(\mathbf { j }\).
At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d \mathrm {~km}\).
- By finding an expression for \(\overrightarrow { P Q }\), show that
$$d ^ { 2 } = 25 t ^ { 2 } - 92 t + 292$$
Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,
- find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer.
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- In taking off, an aircraft moves on a straight runway \(A B\) of length 1.2 km . The aircraft moves from \(A\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
- the acceleration of the aircraft,
- the distance \(B C\).
- Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 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 \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find
- the speed of \(A\) immediately after the collision,
- the magnitude of the impulse exerted on \(B\) in the collision.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-018_282_707_278_699}
\end{figure}
A smooth bead \(B\) is threaded on a light inextensible string. The ends of the string are attached to two fixed points \(A\) and \(C\) on the same horizontal level. The bead is held in equilibrium by a horizontal force of magnitude 6 N acting parallel to \(A C\). The bead \(B\) is vertically below \(C\) and \(\angle B A C = \alpha\), as shown in Figure 1. Given that \(\tan \alpha = \frac { 3 } { 4 }\), find - the tension in the string,
- the weight of the bead.
4.
\begin{figure}[h]
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\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-019_256_615_280_659}
\end{figure}
A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of \(20 ^ { \circ }\) to the horizontal, as shown in Figure 2. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N . The coefficient of friction between the box and the plane is 0.6 . By modelling the box as a particle, find - the normal reaction of the plane on the box,
- the acceleration of the box.
5. A train is travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal track. The driver sees a red signal 135 m ahead and immediately applies the brakes. The train immediately decelerates with constant deceleration for 12 s , reducing its speed to \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The driver then releases the brakes and allows the train to travel at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 15 s . He then applies the brakes again and the train slows down with constant deceleration, coming to rest as it reaches the signal. - Sketch a speed-time graph to show the motion of the train,
- Find the distance travelled by the train from the moment when the brakes are first applied to the moment when its speed first reaches \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Find the total time from the moment when the brakes are first applied to the moment when the train comes to rest.
6.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-022_212_741_287_660}
\end{figure}
A uniform beam \(A B\) has mass 12 kg and length 3 m . The beam rests in equilibrium in a horizontal position, resting on two smooth supports. One support is at the end \(A\), the other at a point \(C\) on the beam, where \(B C = 1 \mathrm {~m}\), as shown in Figure 3. The beam is modelled as a uniform rod. - Find the reaction on the beam at \(C\).
A woman of mass 48 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The reactions on the beam at \(A\) and \(C\) are now equal.
- Find the distance \(A D\).
\includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-023_72_58_2632_1873}
7.
\begin{figure}[h]
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\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-024_206_925_281_511}
\end{figure}
Figure 4 shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of \(15 ^ { \circ }\) to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry's engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find - the acceleration of the lorry and the car,
- the tension in the towbar.
When the speed of the vehicles is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N ,
- find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest.
- State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer.
- \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal vectors due east and north respectively.]
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.
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\section*{Paper Reference(s)}
6677/01
\section*{Edexcel GCE }
Examiner's use only
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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.