Questions M1 (1912 questions)

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Edexcel M1 2004 June Q6
13 marks Standard +0.3
6. A small boat \(S\), drifting in the sea, is modelled as a particle moving in a straight line at constant speed. When first sighted at \(0900 , S\) is at a point with position vector \(( 4 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively. At \(0945 , S\) is at the point with position vector ( \(7 \mathbf { i } - 7.5 \mathbf { j }\) ) km. At time \(t\) hours after 0900, \(S\) is at the point with position vector \(\mathbf { s } \mathrm { km }\).
  1. Calculate the bearing on which \(S\) is drifting.
  2. Find an expression for \(\mathbf { s }\) in terms of \(t\). At 1000 a motor boat \(M\) leaves \(O\) and travels with constant velocity ( \(p \mathbf { i } + q \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Given that \(M\) intercepts \(S\) at 1015,
  3. calculate the value of \(p\) and the value of \(q\).
    (6)
Edexcel M1 2004 June Q7
17 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{57a51cfd-7206-4f34-9744-44255789188d-5_196_1100_363_506}
\end{figure} Two particles \(P\) and \(Q\), of mass 4 kg and 6 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. The coefficient of friction between each particle and the plane is \(\frac { 2 } { 7 }\). A constant force of magnitude 40 N is then applied to \(Q\) in the direction \(P Q\), as shown in Fig. 4.
  1. Show that the acceleration of \(Q\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Calculate the tension in the string when the system is moving.
  3. State how you have used the information that the string is inextensible. After the particles have been moving for 7 s , the string breaks. The particle \(Q\) remains under the action of the force of magnitude 40 N .
  4. Show that \(P\) continues to move for a further 3 seconds.
  5. Calculate the speed of \(Q\) at the instant when \(P\) comes to rest. END
Edexcel M1 2005 June Q1
6 marks Moderate -0.3
  1. 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
    1. the acceleration of the aircraft,
    2. the distance \(B C\).
    3. 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
    4. the speed of \(A\) immediately after the collision,
    5. 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]{bd649c3c-6172-4522-bddc-a6d70088ef89-04_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
  2. the tension in the string,
  3. the weight of the bead.
Edexcel M1 2005 June Q4
8 marks Moderate -0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{bd649c3c-6172-4522-bddc-a6d70088ef89-05_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
  1. the normal reaction of the plane on the box,
  2. the acceleration of the box.
Edexcel M1 2005 June Q5
10 marks Moderate -0.8
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.
  1. Sketch a speed-time graph to show the motion of the train,
  2. Find the distance travelled by the train from the moment when the brakes are first applied to the moment when its speed first reaches \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the total time from the moment when the brakes are first applied to the moment when the train comes to rest.
Edexcel M1 2005 June Q6
10 marks Moderate -0.5
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{bd649c3c-6172-4522-bddc-a6d70088ef89-08_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.
  1. 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.
  2. Find the distance \(A D\).
    \includegraphics[max width=\textwidth, alt={}, center]{bd649c3c-6172-4522-bddc-a6d70088ef89-09_72_58_2632_1873}
Edexcel M1 2005 June Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{bd649c3c-6172-4522-bddc-a6d70088ef89-10_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
  1. the acceleration of the lorry and the car,
  2. 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 ,
  3. find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest.
  4. 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.
Edexcel M1 2005 June Q8
13 marks Moderate -0.3
  1. \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
  1. the speed of the ball,
  2. the position vector of the ball after \(t\) seconds. The point \(B\) on the field has position vector \(( 10 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m }\).
  3. 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,
  4. find the value of \(v\).
  5. 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.
Edexcel M1 2006 June Q1
6 marks Easy -1.3
1. Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{3a8395fd-6e44-48a1-8c97-3365a284956a-02_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
  1. the graph from \(t = 0\) to \(t = 3\) is a straight line,
  2. the graph from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis.
  3. Find the distance travelled by the cyclist during this 7 s period.
Edexcel M1 2006 June Q2
7 marks Moderate -0.8
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
  1. the speed of \(A\) immediately after the collision, stating clearly whether the direction of motion of \(A\) is changed by the collision,
  2. the magnitude of the impulse exerted on \(B\) in the collision, stating clearly the units in which your answer is given.
Edexcel M1 2006 June Q3
10 marks Moderate -0.8
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
  1. the acceleration of the train,
  2. the speed of the front of the train when it passes \(C\),
  3. the time that elapses from the instant the front of the train passes \(B\) to the instant it passes \(C\).
Edexcel M1 2006 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{3a8395fd-6e44-48a1-8c97-3365a284956a-05_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.
  1. Find the coefficient of friction between \(P\) and the plane. The force of magnitude 4 N is removed.
  2. Find the acceleration of \(P\) down the plane.
Edexcel M1 2006 June Q5
13 marks Moderate -0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{3a8395fd-6e44-48a1-8c97-3365a284956a-07_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
  1. the tension in the cable at \(A\),
  2. 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 \(A\).
  3. Find the value of \(W\).
Edexcel M1 2006 June Q6
13 marks Moderate -0.3
  1. 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
    1. the acceleration of the car,
    2. 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,
  2. find the distance moved by the car in the first 4 s after the tow-rope breaks.
    (6)
  3. State how you have used the modelling assumption that the tow-rope is inextensible.
Edexcel M1 2006 June Q7
15 marks Moderate -0.3
  1. \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
  1. the speed of \(S\),
  2. 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.
  3. 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
  4. an expression for the position vector of the ship \(t\) hours after 1400,
  5. the time when \(S\) will be due east of \(R\),
  6. the distance of \(S\) from \(R\) at the time 1600.
Edexcel M1 2007 June Q1
7 marks Moderate -0.8
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5b5d70b1-1eb6-461f-9277-5912b914f443-02_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
  1. the tension in the string,
  2. the weight of \(P\).
Edexcel M1 2007 June Q2
7 marks Moderate -0.3
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
  1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision,
  2. the value of \(m\).
Edexcel M1 2007 June Q3
9 marks Moderate -0.3
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5b5d70b1-1eb6-461f-9277-5912b914f443-04_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.
  1. 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.
  2. Find the distance \(A D\).
Edexcel M1 2007 June Q4
11 marks Moderate -0.8
  1. 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 \(t = 30 \mathrm {~s}\).
    1. 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
  2. the value of \(V\),
  3. the deceleration of the car between \(t = 10 \mathrm {~s}\) and \(t = 18 \mathrm {~s}\).
Edexcel M1 2007 June Q5
10 marks Moderate -0.8
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{5b5d70b1-1eb6-461f-9277-5912b914f443-08_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
  1. the normal reaction between the ring and the rod,
  2. the value of \(\mu\).
Edexcel M1 2007 June Q6
17 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{5b5d70b1-1eb6-461f-9277-5912b914f443-10_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.
  1. Show that the acceleration of \(P\) as it descends is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the tension in the string as \(P\) descends.
  3. Show that \(m = \frac { 5 } { 18 }\).
  4. 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.
  5. Find the time between the instant when \(P\) strikes the ground and the instant when the string becomes taut again.
Edexcel M1 2007 June Q7
14 marks Standard +0.3
  1. 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 \(( 8 \mathbf { i } + 11 \mathbf { j } ) \mathrm { km }\).
    1. 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 \(\mathbf { b } \mathrm { km }\).
  2. 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 \(B\),
  3. find the value of \(\lambda\),
  4. show that, before \(C\) intercepts \(B\), the boats are moving with the same speed.
Edexcel M1 2008 June Q1
6 marks Easy -1.2
  1. 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 \(P Q\).
    1. Find the speed of \(P\) immediately before it collides with \(Q\).
    Immediately after the collision between \(P\) and \(Q\), the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Show that immediately after the collision \(P\) is at rest.
Edexcel M1 2008 June Q2
7 marks Moderate -0.8
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
  1. the value of \(u\),
  2. the value of \(T\).
Edexcel M1 2008 June Q3
8 marks Moderate -0.8
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
  1. the angle between the acceleration and \(\mathbf { i }\),
  2. 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 \(t = 5\).