Questions — Edexcel M1 (663 questions)

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Edexcel M1 2003 June Q4
8 marks Standard +0.3
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
10 marks Moderate -0.3
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
11 marks Moderate -0.3
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
12 marks Standard +0.3
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
13 marks Moderate -0.3
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 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.
[diagram]
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 \(t = 5\).
Edexcel M1 2008 June Q4
9 marks Moderate -0.8
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 \(A B = 1410 \mathrm {~m}\).
  1. Sketch, in the space below, a speed-time graph to show the motion of the car from \(A\) to \(B\).
  2. 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 \(\_\_\_\_\)
Edexcel M1 2008 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-07_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
  1. the magnitude of \(\mathbf { R }\),
  2. the value of \(X\).
Edexcel M1 2008 June Q6
10 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-08_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.
  1. 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 \(B\).
  2. Find the distance of the centre of mass of the plank from \(A\).