Questions — Edexcel (9685 questions)

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Edexcel M1 Q4
10 marks Standard +0.3
4. The force \(\mathbf { F } _ { \mathbf { 1 } } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) acts at the point \(A\) on a lamina where the position vector of \(A\), relative to a fixed origin \(O\), is \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\).
  1. Calculate the magnitude and the sense of the moment of the force about \(O\). Another force \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } )\), acts at the point \(B\) with position vector ( \({ } ^ { - } \mathbf { i } + 4 \mathbf { j }\) ) m so that the resultant moment of the two forces, \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\), about \(O\) is zero. Given also that the moment of \(\mathbf { F } _ { 2 }\) about \(A\) is 34 Ns in a clockwise sense,
  2. find the values of \(p\) and \(q\).
Edexcel M1 Q5
13 marks Moderate -0.3
5. A car and a motorbike are at rest adjacent to one another at a set of traffic lights on a long, straight stretch of road. They set off simultaneously at time \(t = 0\). The motorcyclist accelerates uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) until he reaches a speed of \(30 \mathrm {~ms} ^ { - 1 }\) which he then maintains. The car driver accelerates uniformly for 9 seconds until she reaches \(36 \mathrm {~ms} ^ { - 1 }\) and then remains at this speed.
  1. Find the acceleration of the car.
  2. Draw on the same diagram speed-time graphs to illustrate the movements of both vehicles.
  3. Find the value of \(t\) when the car again draws level with the motorcyclist.
Edexcel M1 Q6
13 marks Standard +0.3
6. Corinne and her brother Dermot are lifted by their parents onto the two ends of a rope which is slung over a large, horizontal branch. When their parents let go of them Dermot, whose mass is 54 kg , begins to descend with an acceleration of \(1 \mathrm {~ms} ^ { - 2 }\). By modelling the children as a pair of particles connected by a light inextensible string, and the branch as a smooth pulley,
  1. show that Corinne's mass is 44 kg ,
  2. calculate the tension in the rope,
  3. find the force on the branch. In a more sophisticated model, the branch is assumed to be rough.
  4. Explain what effect this would have on the initial acceleration of the children.
    (1 mark)
Edexcel M1 Q7
14 marks Standard +0.3
7. Two particles \(A\) and \(B\), of mass \(3 M \mathrm {~kg}\) and \(2 M \mathrm {~kg}\) respectively, are moving towards each other on a rough horizontal track. Just before they collide, \(A\) has speed \(3 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(5 \mathrm {~ms} ^ { - 1 }\). Immediately after the impact, the direction of motion of both particles has been reversed and they are both travelling at the same speed, \(v\).
  1. Show that \(v = 1 \mathrm {~ms} ^ { - 1 }\). The magnitude of the impulse exerted on \(A\) during the collision is 24 Ns.
  2. Find the value of \(M\). Given that the coefficient of friction between \(A\) and the track is 0.1 ,
  3. find the time taken from the moment of impact until \(A\) comes to rest. END
Edexcel M1 Q1
5 marks Moderate -0.8
  1. A particle, \(P\), of mass 5 kg moves with speed \(3 \mathrm {~ms} ^ { - 1 }\) along a smooth horizontal track. It strikes a particle \(Q\) of mass 2 kg which is at rest on the track. Immediately after the collision, \(P\) and \(Q\) move in the same direction with speeds \(v\) and \(2 v \mathrm {~ms} ^ { - 1 }\) respectively.
    1. Calculate the value of \(v\).
    2. Calculate the magnitude of the impulse received by \(Q\) on impact.
    3. A particle \(P\) moves with a constant velocity \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) with respect to a fixed origin \(O\). It passes through the point \(A\) whose position vector is \(( 2 \mathbf { i } + 11 \mathbf { j } )\) m at \(t = 0\).
    4. Find the angle in degrees that the velocity vector of \(P\) makes with the vector \(\mathbf { i }\).
    5. Calculate the distance of \(P\) from \(O\) when \(t = 2\).
    6. A car of mass 1250 kg is moving at constant speed up a hill, inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 10 }\). The driving force produced by the engine is 1800 N .
    7. Calculate the resistance to motion which the car experiences.
    At the top of the hill, the road becomes horizontal.
  2. Find the initial acceleration of the car.
Edexcel M1 Q4
10 marks Standard +0.3
4. A non-uniform plank \(A B\) of mass 20 kg and length 6 m is supported at both ends so that it is horizontal. When a woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\), the magnitude of the reaction at \(A\) is \(35 g \mathrm {~N}\).
  1. Suggest a suitable model for
    1. the plank,
    2. the woman.
  2. Calculate the magnitude of the reaction at \(B\), giving your answer in terms of \(g\).
  3. Explain briefly, in the context of the problem, the term 'non-uniform'.
  4. Find the distance of the centre of mass of the plank from \(A\).
Edexcel M1 Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf4361b8-acd2-4133-81b7-2f68d018486f-3_99_1036_242_379} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The points \(A , O\) and \(B\) lie on a straight horizontal track as shown in Figure 1. \(A\) is 20 m from \(O\) and \(B\) is on the other side of \(O\) at a distance \(x \mathrm {~m}\) from \(O\). At time \(t = 0\), a particle \(P\) starts from rest at \(O\) and moves towards \(B\) with uniform acceleration of \(3 \mathrm {~ms} ^ { - 2 }\). At the same instant, another particle \(Q\), which is at the point \(A\), is moving with a velocity of \(3 \mathrm {~ms} ^ { - 1 }\) in the direction of \(O\) with uniform acceleration of \(4 \mathrm {~ms} ^ { - 2 }\) in the same direction. Given that the \(Q\) collides with \(P\) at \(B\), find the value of \(x\).
Edexcel M1 Q6
11 marks Moderate -0.3
6. A sledge of mass 4 kg rests in limiting equilibrium on a rough slope inclined at an angle \(10 ^ { \circ }\) to the horizontal. By modelling the sledge as a particle,
  1. show that the coefficient of friction, \(\mu\), between the sledge and the ground is 0.176 correct to 3 significant figures.
    (6 marks)
    The sledge is placed on a steeper part of the slope which is inclined at an angle \(30 ^ { \circ }\) to the horizontal. The value of \(\mu\) remains unchanged.
  2. Find the minimum extra force required along the line of greatest slope to prevent the sledge from slipping down the hill.
Edexcel M1 Q7
12 marks Challenging +1.2
7. Whilst looking over the edge of a vertical cliff, 122.5 metres in height, Jim dislodges a stone. The stone falls freely from rest towards the sea below. Ignoring the effect of air resistance,
  1. calculate the time it would take for the stone to reach the sea,
  2. find the speed with which the stone would hit the water. Two seconds after the stone begins to fall, Jim throws a tennis ball downwards at the stone. The tennis ball's initial speed is \(u \mathrm {~ms} ^ { - 1 }\) and it hits the stone before they both reach the water.
  3. Find the minimum value of \(u\).
  4. If you had taken air resistance into account in your calculations, what effect would this have had on your answer to part (c)? Explain your answer.
    (2 marks)
Edexcel M1 Q8
14 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bf4361b8-acd2-4133-81b7-2f68d018486f-4_478_529_1142_589} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, attached to the ends of a light, inextensible string which passes over a smooth, fixed pulley. The system is released from rest with \(P\) and \(Q\) at the same level 1.5 metres above the ground and 2 metres below the pulley.
  1. Show that the initial acceleration of the system is \(\frac { g } { 5 } \mathrm {~ms} ^ { - 2 }\).
  2. Find the tension in the string.
  3. Find the speed with which \(P\) hits the ground. When \(P\) hits the ground, it does not rebound.
  4. What is the closest that \(Q\) gets to the pulley.
Edexcel M1 Q1
4 marks Moderate -0.8
  1. Three forces \(( - 5 \mathbf { i } + 4 p \mathbf { j } ) \mathrm { N } , ( 2 q \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) and \(( \mathbf { i } + \mathbf { j } ) \mathrm { N }\) act on a particle \(A\) of mass 2 kg .
Given that \(A\) is in equilibrium, find the values of \(p\) and \(q\).
Edexcel M1 Q2
7 marks Moderate -0.8
2. An underground train accelerates uniformly from rest at station \(A\) to a velocity of \(24 \mathrm {~ms} ^ { - 1 }\). It maintains this speed for 84 seconds, until it decelerates uniformly to rest at station \(B\). The total journey time is 116 seconds and the magnitudes of the acceleration and deceleration are equal.
  1. Find the time it takes the train to accelerate from rest to \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Illustrate this information on a velocity-time graph.
  3. Using your graph, or otherwise, find the distance between the two stations.
Edexcel M1 Q3
8 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{60b9db45-b48e-40a1-bd22-909e11877bc3-2_442_805_1023_719} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows the forces acting on a particle, \(P\). These consist of a 20 N force to the South, a 6 N force to the East, an 18 N force \(30 ^ { \circ }\) West of North and two unknown forces \(X\) and \(Y\) which act to the North-East and North respectively. Given that \(P\) is in equilibrium,
  1. show that \(X\) has magnitude \(3 \sqrt { } 2 \mathrm {~N}\),
  2. find the exact value of \(Y\).
Edexcel M1 Q4
8 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{60b9db45-b48e-40a1-bd22-909e11877bc3-3_275_842_194_408} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a uniform plank \(A B\) of mass 50 kg and length 5 m which overhangs a river by 2 m . When a boy of mass 20 kg stands at \(A\), his sister can walk to within 0.3 m of \(B\), at which point the plank is in limiting equilibrium.
  1. What is the mass of the girl?
  2. Find the smallest extra weight which must be placed at \(A\) to enable the girl to walk right to the end \(B\).
  3. How have you used the fact that the plank is uniform?
Edexcel M1 Q5
8 marks Moderate -0.3
5. A cricket ball of mass 0.3 kg is approaching a batsman at \({ } ^ { - } 30 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The batsman hits the ball with a 1.5 kg bat moving with velocity \(15 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). Contact between bat and ball lasts for 0.2 seconds. Immediately after this, bat and ball move with velocities \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) and \(v \mathbf { i } \mathrm {~ms} ^ { - 1 }\) respectively.
  1. Suggest a suitable model for the cricket ball.
  2. Calculate the value of \(v\).
  3. Find the magnitude of the force with which the batsman hits the ball.
Edexcel M1 Q6
10 marks Moderate -0.3
6. A boy kicks a football vertically upwards from a height of 0.6 m above the ground with a speed of \(10.5 \mathrm {~ms} ^ { - 1 }\). The ball is modelled as a particle and air resistance is ignored.
  1. Find the greatest height above the ground reached by the ball.
  2. Calculate the length of time for which the ball is more than 2 m above the ground.
Edexcel M1 Q7
11 marks Standard +0.3
7. A particle has an initial velocity of \(( \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and is accelerating uniformly in the direction \(( 2 \mathbf { i } + \mathbf { j } )\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. Given that the magnitude of the acceleration is \(3 \sqrt { } 5 \mathrm {~ms} ^ { - 2 }\),
  1. show that, after \(t\) seconds, the velocity vector of the particle is $$[ ( 6 t + 1 ) \mathbf { i } + ( 3 t - 5 ) \mathbf { j } ] \mathrm { ms } ^ { - 1 }$$
  2. Using your answer to part (a), or otherwise, find the value of \(t\) for which the speed of the particle is at its minimum.
    (5 marks)
Edexcel M1 Q8
19 marks Standard +0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{60b9db45-b48e-40a1-bd22-909e11877bc3-4_442_924_877_443} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows two particles \(A\) and \(B\), of mass \(5 M\) and \(3 M\) respectively, attached to the ends of a light inextensible string of length 4 m . The string passes over a smooth pulley which is fixed to the edge of a rough horizontal table 2 m high. Particle \(A\) lies on the table at a distance of 3 m from the pulley, whilst particle \(B\) hangs freely over the edge of the table 1 m above the ground. The coefficient of friction between \(A\) and the table is \(\frac { 3 } { 20 }\). The system is released from rest with the string taut.
  1. Show that the initial acceleration of the system is \(\frac { 9 } { 32 } \mathrm {~g} \mathrm {~ms} ^ { - 2 }\).
  2. Find, in terms of \(g\), the speed of \(A\) immediately before \(B\) hits the ground. When \(B\) hits the ground, it comes to rest and the string becomes slack.
  3. Calculate how far particle \(A\) is from the pulley when it comes to rest. END
Edexcel M1 Q1
7 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0de1908-cf67-460f-9473-b2dfded95b33-2_257_693_239_447} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a particle \(P\) of mass 4 kg on a smooth plane inclined at \(15 ^ { \circ }\) to the horizontal. \(P\) is held in equilibrium by a horizontal force, \(F\).
  1. Show that the normal reaction exerted by the plane on \(P\) is 40.6 N correct to 3 significant figures.
  2. Calculate the value of \(F\).
Edexcel M1 Q2
7 marks Moderate -0.3
2. During trials of a bullet-proof vest, a shotgun of mass 2 kg is used to fire a bullet of mass 30 g horizontally at the vest. The initial speed of the bullet is \(100 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the initial speed of recoil of the gun. The bullet hits the vest horizontally at a speed of \(80 \mathrm {~ms} ^ { - 1 }\) and is brought uniformly to rest in a distance of 2 cm .
  2. Find the magnitude of the force exerted by the vest on the bullet in bringing it to rest.
    (4 marks)
Edexcel M1 Q3
7 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0de1908-cf67-460f-9473-b2dfded95b33-2_387_460_1626_726} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows 4 points \(A , B , C\) and \(D\) arranged such that they form the corners of a square of side 2 m . Forces of \(5 \mathrm {~N} , 3 \mathrm {~N} , 2 \mathrm {~N}\) and 4 N act in the directions \(\overrightarrow { A B } , \overrightarrow { B C } , \overrightarrow { D C }\) and \(\overrightarrow { D A }\) respectively.
  1. Calculate the magnitude and sense of the resultant moment about \(A\). An additional force of magnitude \(X\) Newtons is added in the direction \(\overrightarrow { C A }\). The resultant moment of all the forces about \(D\) is now zero.
  2. Find, in the form \(k \sqrt { } 2\), the value of \(X\).
Edexcel M1 Q4
11 marks Standard +0.3
4. A lift of mass 70 kg is supported by a cable which remains taut at all times. A man of mass 90 kg gets into the lift and it begins to descend vertically from rest with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate, giving your answers correct to 3 significant figures,
  1. the magnitude of the force which the lift exerts on the man,
  2. the tension in the cable. Prior to slowing down, the lift is moving at \(2 \mathrm {~ms} ^ { - 1 }\). It then uniformly decelerates until it is brought to rest.
  3. Find the impulse exerted by the cable on the lift in bringing the lift to rest.
  4. Given that it takes 2 seconds to come to rest, use your answer to part (c) to calculate the magnitude of the force exerted by the cable on the lift in bringing the lift to rest.
    (2 marks)
Edexcel M1 Q5
11 marks Moderate -0.3
5. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively. At midday a motor boat \(A\) is 6 km east of a fixed origin \(O\) and is moving with constant velocity ( \({ } ^ { - } 4 \mathbf { i } + \mathbf { j }\) ) \(\mathrm { km } \mathrm { h } ^ { - 1 }\). At the same time, another boat \(B\) is 3 km north of \(O\) and is moving with uniform velocity \(( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Show that, at time \(T\) hours after midday, the position vector of \(A\) is \([ ( 6 - 4 T ) \mathbf { i } + T \mathbf { j } ] \mathrm { km }\) and find a similar expression for the position vector of \(B\) at this time.
  2. Hence show that, at time \(T\), the position vector of \(B\) relative to \(A\) is $$[ ( 8 T - 6 ) \mathbf { i } + ( 3 - 4 T ) \mathbf { j } ] \mathrm { km }$$
  3. By using your answer to part (b), or otherwise, show that the boats would collide if they continued at the same velocities and find the time at which the collision would occur.
Edexcel M1 Q6
13 marks Standard +0.3
6. A student attempts to sketch the acceleration-time graph of a parachutist who jumps from a plane at a height of 2200 m above the ground. The student assumes that the parachutist falls freely from rest under gravity until she is 240 m from the ground at which point she opens her parachute. The student makes the assumption that, at this point, the velocity of the parachutist is immediately reduced to a value which remains constant until she reaches the ground 140 seconds after she left the plane. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0de1908-cf67-460f-9473-b2dfded95b33-4_314_1013_598_383} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} The student decides to ignore air resistance and his sketch is shown in Figure 3. The value \(t _ { 1 }\) is used by the student to denote the time at which the parachute is opened. Using the model proposed by the student, calculate
  1. the speed of the parachutist immediately before she opens her parachute,
  2. the value of \(t _ { 1 }\),
  3. the speed of the parachutist after the parachute is opened.
  4. Comment on two features of the student's model which are unrealistic and say what effect taking account of these would have had on the values which you calculated in parts (a) and (b).
    (4 marks)
Edexcel M1 Q7
19 marks Standard +0.3
7. A machine fires ball-bearings up the line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The coefficient of friction between the ball-bearings and the plane is \(\frac { 1 } { 4 }\).
  1. Show that the magnitude of the acceleration of the ball-bearings is \(\frac { 4 } { 5 } g\) and state its direction. Given that the machine is placed at a point \(A , 30 \mathrm {~m}\) from the top edge of the plane, and the ball-bearings are projected with an initial speed of \(20 \mathrm {~ms} ^ { - 1 }\),
  2. find, giving your answer to the nearest cm , how close the ball-bearings get to the top edge of the plane.
  3. How long does it take for a ball-bearing to travel from the highest point it reaches back down to the point \(A\) again? END