Questions — Edexcel M1 (599 questions)

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Edexcel M1 2023 October Q3
  1. A hammer is used to hit a tent peg into soft ground.
The hammer has mass 1.8 kg and the tent peg has mass 0.2 kg .
The hammer and tent peg are both modelled as particles and the impact is modelled as a direct collision. Immediately before the impact, the tent peg is stationary and the hammer is moving vertically downwards with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Immediately after the impact, the hammer and tent peg move together, vertically downwards, with the same speed \(v \mathrm {~ms} ^ { - 1 }\)
  1. Find the value of \(v\)
  2. Find the magnitude of the impulse exerted on the tent peg by the hammer, stating the units of your answer. The ground exerts a constant vertical resistive force of magnitude \(R\) newtons, bringing the hammer and tent peg to rest after they travel a distance of 12 cm .
  3. Find the value of \(R\).
Edexcel M1 2023 October Q4
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively.]
A particle \(P\) moves with constant acceleration \(( - \lambda \mathbf { i } + 2 \lambda \mathbf { j } ) \mathrm { ms } ^ { - 2 }\), where \(\lambda\) is a positive constant. At time \(t = 0\), the velocity of \(P\) is \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
  1. Find the velocity of \(P\) when \(t = 5 \mathrm {~s}\), giving your answer in terms of \(\mathbf { i } , \mathbf { j }\) and \(\lambda\). The speed of \(P\) when \(t = 5 \mathrm {~s}\) is \(13 \mathrm {~ms} ^ { - 1 }\)
  2. Show that $$25 \lambda ^ { 2 } - 42 \lambda - 16 = 0$$
  3. Find the direction of motion of \(P\) when \(t = 4 \mathrm {~s}\), giving your answer as a bearing to the nearest degree.
Edexcel M1 2023 October Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{017cc2b0-9ec3-45ff-94c0-9d989badfd5d-16_757_460_246_804} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small ring of mass 0.2 kg is attached to one end of a light inextensible string.
The ring is threaded onto a fixed rough vertical rod.
The string is taut and makes an angle \(\theta\) with the rod, as shown in Figure 3, where \(\tan \theta = \frac { 12 } { 5 }\) Given that the ring is in equilibrium and that the tension in the string is 10 N ,
  1. find the magnitude of the frictional force acting on the ring,
  2. state the direction of the frictional force acting on the ring. The coefficient of friction between the ring and the rod is \(\frac { 1 } { 4 }\)
    Given that the ring is in equilibrium, and that the tension in the string, \(T\) newtons, can now vary,
    1. find the minimum value of \(T\)
    2. find the maximum value of \(T\)
Edexcel M1 2023 October Q6
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]
At 12:00, a ship \(P\) sets sail from a harbour with position vector \(( 15 \mathbf { i } + 36 \mathbf { j } ) \mathrm { km }\). At 12:30, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). Given that \(P\) moves with constant velocity,
  1. show that the velocity of \(P\) is \(( 10 \mathbf { i } - 4 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) At time \(t\) hours after 12:00, the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\).
  2. Find an expression for \(\mathbf { p }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(t\). A second ship \(Q\) is also travelling at a constant velocity.
    At time \(t\) hours after 12:00, the position vector of \(Q\) is given by \(\mathbf { q } \mathrm { km }\), where $$\mathbf { q } = ( 42 - 8 t ) \mathbf { i } + ( 9 + 14 t ) \mathbf { j }$$ Ships \(P\) and \(Q\) are modelled as particles.
    If both ships maintained their course,
    1. verify that they would collide at 13:30
    2. find the position vector of the point at which the collision would occur. At 12:30 \(Q\) changes speed and direction to avoid the collision.
      Ship \(Q\) now travels due north with a constant speed of \(15 \mathrm { kmh } ^ { - 1 }\)
      Ship \(P\) maintains the same constant velocity throughout.
  4. Find the exact distance between \(P\) and \(Q\) at 14:30
Edexcel M1 2023 October Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{017cc2b0-9ec3-45ff-94c0-9d989badfd5d-24_339_942_244_635} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a block \(A\) of mass \(m\) held at rest on a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal and the coefficient of friction between the block and the plane is \(\mu\). One end of a light inextensible string is now attached to \(A\). The string passes over a small smooth pulley which is fixed at the top of the plane.
The other end of the string is attached to a block \(B\) of mass \(k m\).
Block \(B\) hangs vertically below the pulley, with the string taut.
The string from \(A\) to the pulley lies along a line of greatest slope of the plane.
Both \(A\) and \(B\) are modelled as particles.
When the system is released from rest, \(A\) moves up the plane and the tension in the string is \(\frac { 4 m g } { 3 }\) Given that \(\mu = \frac { 1 } { 3 }\) and \(\tan \alpha = \frac { 7 } { 24 }\)
    1. find the magnitude of the acceleration of \(A\), giving your answer in terms of \(g\),
    2. find the value of \(k\).
  2. Find the magnitude of the resultant force exerted on the pulley by the string, giving your answer in terms of \(m\) and \(g\).
Edexcel M1 2018 Specimen Q2
2. Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(k m\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2 u\). As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.
  1. Find the value of \(k\).
  2. Find, in terms of \(m\) and \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision.
Edexcel M1 2018 Specimen Q3
3. A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The blocks are modelled as particles.
  1. Find the value of \(h\). Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
  2. find the value of \(R\).
    \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-07_2252_51_315_36}
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
    \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-09_2249_45_318_37}
Edexcel M1 2018 Specimen Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6ab8838f-d6f8-4761-8def-1022d97d4e82-10_238_1161_267_388} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A diving board \(A B\) consists of a wooden plank of length 4 m and mass 30 kg . The plank is held at rest in a horizontal position by two supports at the points \(A\) and \(C\), where \(A C = 0.6 \mathrm {~m}\), as shown in Figure 1. The force on the plank at \(A\) acts vertically downwards and the force on the plank at \(C\) acts vertically upwards. A diver of mass 50 kg is standing on the board at the end \(B\). The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium.
  1. Find
    1. the magnitude of the force acting on the plank at \(A\),
    2. the magnitude of the force acting on the plank at \(C\). The support at \(A\) will break if subjected to a force whose magnitude is greater than 5000 N .
  2. Find, in kg, the greatest integer mass of a diver who can stand on the board at \(B\) without breaking the support at \(A\).
  3. Explain how you have used the fact that the diver is modelled as a particle.
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
Edexcel M1 2018 Specimen Q5
  1. Two forces, \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\), act on a particle \(A\).
    \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\), where \(p\) and \(q\) are constants.
    Given that the resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is parallel to ( \(\mathbf { i } + 2 \mathbf { j }\) ),
    1. show that \(2 p - q + 7 = 0\)
    Given that \(q = 11\) and that the mass of \(A\) is 2 kg , and that \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on \(A\),
  2. find the magnitude of the acceleration of \(A\).
    \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-15_2255_51_314_36}
Edexcel M1 2018 Specimen Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6ab8838f-d6f8-4761-8def-1022d97d4e82-16_264_997_269_461} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two cars, \(A\) and \(B\), move on parallel straight horizontal tracks. Initially \(A\) and \(B\) are both at rest with \(A\) at the point \(P\) and \(B\) at the point \(Q\), as shown in Figure 2. At time \(t = 0\) seconds, \(A\) starts to move with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 3.5 s , reaching a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Car \(A\) then moves with constant speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(a\). Car \(B\) also starts to move at time \(t = 0\) seconds, in the same direction as car \(A\). Car \(B\) moves with a constant acceleration of \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = T\) seconds, \(B\) overtakes \(A\). At this instant \(A\) is moving with constant speed.
  2. On a diagram, sketch, on the same axes, a speed-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and a speed-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\).
  3. Find the value of \(T\).
  4. Find the distance of car \(B\) from the point \(Q\) when \(B\) overtakes \(A\).
  5. On a new diagram, sketch, on the same axes, an acceleration-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and an acceleration-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JIIIM ION OC
    VJYV SIHI NI JLIYM ION OC
Edexcel M1 2018 Specimen Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6ab8838f-d6f8-4761-8def-1022d97d4e82-20_568_1045_264_461} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) and the second plane is inclined to the horizontal at an angle \(\beta\), where \(\tan \beta = \frac { 4 } { 3 }\). Particle \(P\) is on the first plane and particle \(Q\) is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The second plane is smooth. The system is in limiting equilibrium. Given that \(P\) is on the point of slipping down the first plane,
  1. find the value of \(m\),
  2. find the magnitude of the force exerted on the pulley by the string,
  3. find the direction of the force exerted on the pulley by the string. \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-21_2258_50_314_37}
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
    \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-23_2258_50_314_37}
    \includegraphics[max width=\textwidth, alt={}]{6ab8838f-d6f8-4761-8def-1022d97d4e82-24_2655_1830_105_121}
    VIAV SIHI NI JIIYM IONOOVI4V SIHI NI IIIIMM ION OOVEYV SIHI NI JLIYM ION OC
Edexcel M1 2001 January Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{49169cc3-c353-430f-80ce-e14ae7fcd6ea-2_259_792_345_642} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A uniform \(\operatorname { rod } A B\) has weight 70 N and length 3 m . It rests in a horizontal position on two smooth supports placed at \(P\) and \(Q\), where \(A P = 0.5 \mathrm {~m}\), as shown in Fig. 1 . The reaction on the rod at \(P\) has magnitude 20 N . Find
  1. the magnitude of the reaction on the rod at \(Q\),
  2. the distance \(A Q\).
Edexcel M1 2001 January Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{49169cc3-c353-430f-80ce-e14ae7fcd6ea-2_293_725_1267_666} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A particle \(P\) of mass 2 kg is held in equilibrium under gravity by two light inextensible strings. One string is horizontal and the other is inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 2. The tension in the horizontal string is 15 N . The tension in the other string is \(T\) newtons.
  1. Find the size of the angle \(\alpha\).
  2. Find the value of \(T\).
Edexcel M1 2001 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{49169cc3-c353-430f-80ce-e14ae7fcd6ea-3_437_646_305_706} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Two particles \(A\) and \(B\) have masses \(3 m\) and \(k m\) respectively, where \(k > 3\). They are connected by a light inextensible string which passes over a smooth fixed pulley. The system is released from rest with the string taut and the hanging parts of the string vertical, as shown in Fig. 3. While the particles are moving freely, \(A\) has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
  1. Find, in terms of \(m\) and g , the tension in the string.
  2. State why \(B\) also has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
  3. Find the value of \(k\).
  4. State how you have used the fact that the string is light.
Edexcel M1 2001 January Q4
4. A particle \(P\) moves in a straight line with constant velocity. Initially \(P\) is at the point \(A\) with position vector \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m }\) relative to a fixed origin \(O\), and 2 s later it is at the point \(B\) with position vector \(( 6 \mathbf { i } + \mathbf { j } ) \mathrm { m }\).
  1. Find the velocity of \(P\).
  2. Find, in degrees to one decimal place, the size of the angle between the direction of motion of \(P\) and the vector \(\mathbf { i }\).
    (2 marks)
    Three seconds after it passes \(B\) the particle \(P\) reaches the point \(C\).
  3. Find, in m to one decimal place, the distance \(O C\).
Edexcel M1 2001 January Q5
5. Two small balls \(A\) and \(B\) have masses 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, \(A\) and \(B\) move in the same direction and the speed of \(B\) is twice the speed of \(A\). By modelling the balls as particles, find
  1. the speed of \(B\) immediately after the collision,
  2. the magnitude of the impulse exerted on \(B\) in the collision, stating the units in which your answer is given. The table is rough. After the collision, \(B\) moves a distance of 2 m on the table before coming to rest.
  3. Find the coefficient of friction between \(B\) and the table.
Edexcel M1 2001 January Q6
6. A parachutist drops from a helicopter \(H\) and falls vertically from rest towards the ground. Her parachute opens 2 s after she leaves \(H\) and her speed then reduces to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the first 2 s her motion is modelled as that of a particle falling freely under gravity. For the next 5 s the model is motion with constant deceleration, so that her speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of this period. For the rest of the time before she reaches the ground, the model is motion with constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Sketch a speed-time graph to illustrate her motion from \(H\) to the ground.
  2. Find her speed when the parachute opens. A safety rule states that the helicopter must be high enough to allow the parachute to open and for the speed of a parachutist to reduce to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before reaching the ground. Using the assumptions made in the above model,
  3. find the minimum height of \(H\) for which the woman can make a drop without breaking this safety rule. Given that \(H\) is 125 m above the ground when the woman starts her drop,
  4. find the total time taken for her to reach the ground.
  5. State one way in which the model could be refined to make it more realistic.
    (1 mark)
Edexcel M1 2001 January Q7
7. A sledge of mass 78 kg is pulled up a slope by means of a rope. The slope is modelled as a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\). The rope is modelled as light and inextensible and is in a line of greatest slope of the plane. The coefficient of friction between the sledge and the slope is 0.25 . Given that the sledge is accelerating up the slope with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
  1. find the tension in the rope. The rope suddenly breaks. Subsequently the sledge comes to instantaneous rest and then starts sliding down the slope.
  2. Find the acceleration of the sledge down the slope after it has come to instantaneous rest.
    (6 marks)
    END
Edexcel M1 2003 January Q1
  1. A railway truck \(P\) of mass 2000 kg is moving along a straight horizontal track with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The truck \(P\) collides with a truck \(Q\) of mass 3000 kg , which is at rest on the same track. Immediately after the collision \(Q\) moves with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate
    1. the speed of \(P\) immediately after the collision,
    2. the magnitude of the impulse exerted by \(P\) on \(Q\) during the collision.
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{42695f57-bf7a-4cf7-b05f-f76b0a85d82d-2_1009_1169_861_466}
    \end{figure} In Fig. 1, \(\angle A O C = 90 ^ { \circ }\) and \(\angle B O C = \theta ^ { \circ }\). A particle at \(O\) is in equilibrium under the action of three coplanar forces. The three forces have magnitude \(8 \mathrm {~N} , 12 \mathrm {~N}\) and \(X \mathrm {~N}\) and act along \(O A\), \(O B\) and \(O C\) respectively. Calculate
  2. the value, to one decimal place, of \(\theta\),
  3. the value, to 2 decimal places, of \(X\).
Edexcel M1 2003 January Q3
3. A particle \(P\) of mass 0.4 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. Initially the velocity of \(P\) is \(( 6 \mathbf { i } - 27 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and 4 s later the velocity of \(P\) is \(( - 14 \mathbf { i } + 21 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the acceleration of \(P\).
  2. Calculate the magnitude of \(\mathbf { F }\).
Edexcel M1 2003 January Q4
4. Two ships \(P\) and \(Q\) are moving along straight lines with constant velocities. Initially \(P\) is at a point \(O\) and the position vector of \(Q\) relative to \(O\) is ( \(6 \mathbf { i } + 12 \mathbf { j }\) ) km , where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors directed due east and due north respectively. The ship \(P\) is moving with velocity \(10 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and \(Q\) is moving with velocity \(( - 8 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time \(t\) hours the position vectors of \(P\) and \(Q\) relative to \(O\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively.
  1. Find \(\mathbf { p }\) and \(\mathbf { q }\) in terms of \(t\).
  2. Calculate the distance of \(Q\) from \(P\) when \(t = 3\).
  3. Calculate the value of \(t\) when \(Q\) is due north of \(P\).
Edexcel M1 2003 January Q5
5.
\includegraphics[max width=\textwidth, alt={}, center]{42695f57-bf7a-4cf7-b05f-f76b0a85d82d-3_517_745_1387_792} A box of mass 1.5 kg is placed on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the box and plane is \(\frac { 1 } { 3 }\). The box is kept in equilibrium by a light string which lies in a vertical plane containing a line of greatest slope of the plane. The string makes an angle of \(20 ^ { \circ }\) with the plane, as shown in Fig. 2. The box is in limiting equilibrium and is about to move up the plane. The tension in the string is \(T\) newtons. The box is modelled as a particle. Find the value of \(T\).
Edexcel M1 2003 January Q6
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{42695f57-bf7a-4cf7-b05f-f76b0a85d82d-4_293_1165_352_464}
\end{figure} A uniform rod \(A B\) has length 3 m and weight 120 N . The rod rests in equilibrium in a horizontal position, smoothly supported at points \(C\) and \(D\), where \(A C = 0.5 \mathrm {~m}\) and \(A D = 2 \mathrm {~m}\), as shown in Fig. 3. A particle of weight \(W\) newtons is attached to the rod at a point \(E\) where \(A E = x\) metres. The rod remains in equilibrium and the magnitude of the reaction at \(C\) is now twice the magnitude of the reaction at \(D\).
  1. Show that \(W = \frac { 60 } { 1 - x }\).
  2. Hence deduce the range of possible values of \(x\).
Edexcel M1 2003 January Q7
7. A ball is projected vertically upwards with a speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 1.5 m above the ground. The ball moves freely under gravity until it reaches the ground. The greatest height attained by the ball is 25.6 m above \(A\).
  1. Show that \(u = 22.4\). The ball reaches the ground \(T\) seconds after it has been projected from \(A\).
  2. Find, to 2 decimal places, the value of \(T\). The ground is soft and the ball sinks 2.5 cm into the ground before coming to rest. The mass of the ball is 0.6 kg . The ground is assumed to exert a constant resistive force of magnitude \(F\) newtons.
  3. Find, to 3 significant figures, the value of \(F\).
  4. State one physical factor which could be taken into account to make the model used in this question more realistic.
Edexcel M1 2003 January Q8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{42695f57-bf7a-4cf7-b05f-f76b0a85d82d-5_646_1262_485_386}
\end{figure} A particle \(A\) of mass 0.8 kg rests on a 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 particle \(B\) of mass 1.2 kg which hangs freely below the pulley, as shown in Fig. 4. The system is released from rest with the string taut and with \(B\) at a height of 0.6 m above the ground. In the subsequent motion \(A\) does not reach \(P\) before \(B\) reaches the ground. In an initial model of the situation, the table is assumed to be smooth. Using this model, find
  1. the tension in the string before \(B\) reaches the ground,
  2. the time taken by \(B\) to reach the ground. In a refinement of the model, it is assumed that the table is rough and that the coefficient of friction between \(A\) and the table is \(\frac { 1 } { 5 }\). Using this refined model,
  3. find the time taken by \(B\) to reach the ground. \section*{END}