Questions — Edexcel M1 (599 questions)

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Edexcel M1 2001 June Q6
6. A breakdown van of mass 2000 kg is towing a car of mass 1200 kg along a straight horizontal road. The two vehicles are joined by a tow bar which remains parallel to the road. The van and the car experience constant resistances to motion of magnitudes 800 N and 240 N respectively. There is a constant driving force acting on the van of 2320 N . Find
  1. the magnitude of the acceleration of the van and the car,
  2. the tension in the tow bar. The two vehicles come to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). The driving force and the resistances to the motion are unchanged.
  3. Find the magnitude of the acceleration of the van and the car as they move up the hill and state whether their speed increases or decreases.
Edexcel M1 2001 June Q7
7. [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and north respectively] A mountain rescue post \(O\) receives a distress call via a mobile phone from a walker who has broken a leg and cannot move. The walker says he is by a pipeline and he can also see a radio mast which he believes to be south-west of him. The pipeline is known to run north-south for a long distance through the point with position vector \(6 \mathbf { i } \mathrm {~km}\), relative to \(O\). The radio mast is known to be at the point with position vector \(2 \mathbf { j } \mathrm {~km}\), relative to \(O\).
  1. Using the information supplied by the walker, write down his position vector in the form \(( a \mathbf { i } + b \mathbf { j } )\). The rescue party moves at a horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The leader of the party wants to give the walker and idea of how long it will take to for the rescue party to arrive.
  2. Calculate how long it will take for the rescue party to reach the walker's estimated position. The rescue party sets out and walks straight towards the walker's estimated position at a constant horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). After the party has travelled for one hour, the walker rings again. He is very apologetic and says that he now realises that the radio mask is in fact north-west of his position
  3. Find the position vector of the walker.
  4. Find in degrees to one decimal place, the bearing on which the rescue party should now travel in order to reach the walker directly. \section*{END}
Edexcel M1 2003 June Q1
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d91990b5-b7ea-485c-aa4e-fe42b61ca7f8-2_302_807_379_603}
\end{figure} A uniform plank \(A B\) has mass 40 kg and length 4 m . It is supported in a horizontal position by two smooth pivots, one at the end \(A\), the other at the point \(C\) of the plank where \(A C = 3 \mathrm {~m}\), as shown in Fig. 1. A man of mass 80 kg stands on the plank which remains in equilibrium. The magnitudes of the reactions at the two pivots are each equal to \(R\) newtons. By modelling the plank as a rod and the man as a particle, find
  1. the value of \(R\),
  2. the distance of the man from \(A\).
    (4)
Edexcel M1 2003 June Q2
2. Two particles \(A\) and \(B\) have mass 0.12 kg and 0.08 kg respectively. They are initially at rest on a smooth horizontal table. Particle \(A\) is then given an impulse in the direction \(A B\) so that it moves with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards \(B\).
  1. Find the magnitude of this impulse, stating clearly the units in which your answer is given.
    (2) Immediately after the particles collide, the speed of \(A\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its direction of motion being unchanged.
  2. Find the speed of \(B\) immediately after the collision.
  3. Find the magnitude of the impulse exerted on \(A\) in the collision.
Edexcel M1 2003 June Q3
3. A competitor makes a dive from a high springboard into a diving pool. She leaves the springboard vertically with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards. When she leaves the springboard, she is 5 m above the surface of the pool. The diver is modelled as a particle moving vertically under gravity alone and it is assumed that she does not hit the springboard as she descends. Find
  1. her speed when she reaches the surface of the pool,
  2. the time taken to reach the surface of the pool.
  3. State two physical factors which have been ignored in the model.
Edexcel M1 2003 June Q4
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d91990b5-b7ea-485c-aa4e-fe42b61ca7f8-3_355_759_1087_605}
\end{figure} A parcel of mass 5 kg lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The parcel is held in equilibrium by the action of a horizontal force of magnitude 20 N , as shown in Fig. 2. The force acts in a vertical plane through a line of greatest slope of the plane. The parcel is on the point of sliding down the plane. Find the coefficient of friction between the parcel and the plane.
(8)
Edexcel M1 2003 June Q5
5. A particle \(P\) moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t\) seconds, its velocity is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = - 2 \mathbf { i } + 7 \mathbf { j }\).
  1. Find the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf { i }\).
  2. Find the speed of \(P\) when \(t = 3\).
  3. Find the angle between the vector \(\mathbf { j }\) and the direction of motion of \(P\) when \(t = 3\).
Edexcel M1 2003 June Q6
6. A particle \(P\) of mass 3 kg is projected up a line of greatest slope of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The initial speed of P is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the frictional force acting on \(P\) as it moves up the plane,
  2. the distance moved by \(P\) up the plane before \(P\) comes to instantaneous rest.
Edexcel M1 2003 June Q7
7. Two trains \(A\) and \(B\) run on parallel straight tracks. Initially both are at rest in a station and level with each other. At time \(t = 0 , A\) starts to move. It moves with constant acceleration for 12 s up to a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and then moves at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) starts to move in the same direction as \(A\) when \(t = 40\), where \(t\) is measured in seconds. It accelerates with the same initial acceleration as \(A\), up to a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves at a constant speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) overtakes \(A\) after both trains have reached their maximum speed. Train \(B\) overtakes \(A\) when \(t = T\).
  1. Sketch, on the same diagram, the speed-time graphs of both trains for \(0 \leq t \leq T\).
  2. Find the value of \(T\).
Edexcel M1 2003 June Q8
8. A car which has run out of petrol is being towed by a breakdown truck along a straight horizontal road. The truck has mass 1200 kg and the car has mass 800 kg . The truck is connected to the car by a horizontal rope which is modelled as light and inextensible. The truck's engine provides a constant driving force of 2400 N . The resistances to motion of the truck and the car are modelled as constant and of magnitude 600 N and 400 N respectively. Find
  1. the acceleration of the truck and the ear,
  2. the tension in the rope. When the truck and car are moving at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The engine of the truck provides the same driving force as before. The magnitude of the resistance to the motion of the truck remains 600 N .
  3. Show that the truck reaches a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) approximately 6 s earlier than it would have done if the rope had not broken. \section*{END}
Edexcel M1 2004 June Q1
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{57a51cfd-7206-4f34-9744-44255789188d-2_467_1094_352_511}
\end{figure} A particle of weight \(W\) newtons is attached at \(C\) to the ends of two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to two 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 Fig.1. Given the tension in \(A C\) is 50 N , calculate
  1. the tension in \(B C\), to 3 significant figures,
  2. the value of \(W\).
Edexcel M1 2004 June Q2
2. A particle \(P\) is moving with constant acceleration along a straight horizontal line \(A B C\), where \(A C = 24 \mathrm {~m}\). Initially \(P\) is at \(A\) and is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(A B\). After 1.5 s , the direction of motion of \(P\) is unchanged and \(P\) is at \(B\) with speed \(9.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the speed of \(P\) at \(C\) is \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(P\) is 2 kg . When \(P\) reaches \(C\), an impulse of magnitude 30 Ns is applied to \(P\) in the direction \(C B\).
  2. Find the velocity of \(P\) immediately after the impulse has been applied, stating clearly the direction of motion of \(P\) at this instant.
    (3)
Edexcel M1 2004 June Q3
3. A particle \(P\) of mass 2 kg is moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass 4 kg which is at rest on the same horizontal plane. Immediately after the collision, \(P\) and \(Q\) are moving in opposite directions and the speed of \(P\) is one-third the speed of \(Q\).
  1. Show that the speed of \(P\) immediately after the collision is \(\frac { 1 } { 5 } u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision \(P\) continues to move in the same straight line and is brought to rest by a constant resistive force of magnitude 10 N . The distance between the point of collision and the point where \(P\) comes to rest is 1.6 m .
  2. Calculate the value of \(u\).
    (5)
Edexcel M1 2004 June Q4
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{57a51cfd-7206-4f34-9744-44255789188d-3_360_1305_1151_416}
\end{figure} A plank \(A E\), of length 6 m and mass 10 kg , rests in a horizontal position on supports at \(B\) and \(D\), where \(A B = 1 \mathrm {~m}\) and \(D E = 2 \mathrm {~m}\). A child of mass 20 kg stands at \(C\), the mid-point of \(B D\), as shown in Fig. 2. The child is modelled as a particle and the plank as a uniform rod. The child and the plank are in equilibrium. Calculate
  1. the magnitude of the force exerted by the support on the plank at \(B\),
  2. the magnitude of the force exerted by the support on the plank at \(D\). The child now stands at a point \(F\) on the plank. The plank is in equilibrium and on the point of tilting about \(D\).
  3. Calculate the distance \(D F\). \section*{5.} \section*{Figure 3}
    \includegraphics[max width=\textwidth, alt={}]{57a51cfd-7206-4f34-9744-44255789188d-4_422_1142_382_455}
    Figure 3 shows a boat \(B\) of mass 400 kg held at rest on a slipway by a rope. The boat is modelled as a particle and the slipway as a rough plane inclined at \(15 ^ { \circ }\) to the horizontal. The coefficient of friction between \(B\) and the slipway is 0.2 . The rope is modelled as a light, inextensible string, parallel to a line of greatest slope of the plane. The boat is in equilibrium and on the point of sliding down the slipway.
  4. Calculate the tension in the rope.
    (6) The boat is 50 m from the bottom of the slipway. The rope is detached from the boat and the boat slides down the slipway.
  5. Calculate the time taken for the boat to slide to the bottom of the slipway.
    (6)
Edexcel M1 2004 June Q6
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
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
  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
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
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
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
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
  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
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
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
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 \(C\).