Questions M1 (2067 questions)

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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 \(A\).
Edexcel M1 2008 June Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-10_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
  1. the magnitude of the normal reaction of the plane on the package,
  2. the coefficient of friction between the plane and the package.
Edexcel M1 2008 June Q8
15 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9dbbbc01-fb66-460d-a42e-2c37ec8b451a-12_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
  1. the acceleration of \(Q\),
  2. the value of \(\mu\),
  3. the tension in the string.
  4. 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.
  5. Find the time between the instant that the force is removed and the instant that \(Q\) comes to rest.
Edexcel M1 2012 June Q1
6 marks Moderate -0.8
  1. Two particles \(A\) and \(B\), of mass \(5 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are moving in opposite directions along the same straight horizontal line. The particles collide directly. Immediately before the collision, the speeds of \(A\) and \(B\) are \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, the speed of \(A\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find the speed of \(B\) immediately after the collision.
    In the collision, the magnitude of the impulse exerted on \(A\) by \(B\) is 3.3 N s .
  2. Find the value of \(m\).
Edexcel M1 2012 June Q2
7 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c908e75-73df-46be-93bb-09dba2cb3b7e-03_215_716_233_614} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A non-uniform rod \(A B\) has length 3 m and mass 4.5 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(P\) and at \(Q\), where \(A P = 0.8 \mathrm {~m}\) and \(Q B = 0.6 \mathrm {~m}\), as shown in Figure 1. The centre of mass of the rod is at \(G\). Given that the magnitude of the reaction of the support at \(P\) on the rod is twice the magnitude of the reaction of the support at \(Q\) on the rod, find
  1. the magnitude of the reaction of the support at \(Q\) on the rod,
  2. the distance \(A G\).
Edexcel M1 2012 June Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c908e75-73df-46be-93bb-09dba2cb3b7e-04_432_780_210_584} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A box of mass 5 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The box is held in equilibrium by a horizontal force of magnitude 20 N , as shown in Figure 2. The force acts in a vertical plane containing a line of greatest slope of the inclined plane.
The box is in equilibrium and on the point of moving down the plane. The box is modelled as a particle. Find
  1. the magnitude of the normal reaction of the plane on the box,
  2. the coefficient of friction between the box and the plane.
Edexcel M1 2012 June Q4
13 marks Moderate -0.8
  1. A car is moving on a straight horizontal road. At time \(t = 0\), the car is moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is at the point \(A\). The car maintains the speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 25 s . The car then moves with constant deceleration \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reducing its speed from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then moves with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 60 s . The car then moves with constant acceleration until it is moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\).
    1. Sketch a speed-time graph to represent the motion of the car from \(A\) to \(B\).
    2. Find the time for which the car is decelerating.
    Given that the distance from \(A\) to \(B\) is 1960 m ,
  2. find the time taken for the car to move from \(A\) to \(B\).
Edexcel M1 2012 June Q5
12 marks Standard +0.3
  1. A particle \(P\) is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(A\) is 17.5 m above horizontal ground. The particle \(P\) moves freely under gravity until it reaches the ground with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that \(u = 21\)
    At time \(t\) seconds after projection, \(P\) is 19 m above \(A\).
  2. Find the possible values of \(t\). The ground is soft and, after \(P\) reaches the ground, \(P\) sinks vertically downwards into the ground before coming to rest. The mass of \(P\) is 4 kg and the ground is assumed to exert a constant resistive force of magnitude 5000 N on \(P\).
  3. Find the vertical distance that \(P\) sinks into the ground before coming to rest.
Edexcel M1 2012 June Q6
13 marks Moderate -0.8
6. [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 with constant velocity \(( - 12 \mathbf { i } + 7.5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Find the direction in which \(S\) is moving, giving your answer as a bearing. At time \(t\) hours after noon, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\). When \(t = 0 , \mathbf { s } = 40 \mathbf { i } - 6 \mathbf { j }\).
  2. Write down \(\mathbf { s }\) in terms of \(t\). A fixed beacon \(B\) is at the point with position vector \(( 7 \mathbf { i } + 12.5 \mathbf { j } ) \mathrm { km }\).
  3. Find the distance of \(S\) from \(B\) when \(t = 3\)
  4. Find the distance of \(S\) from \(B\) when \(S\) is due north of \(B\).