Questions — Edexcel M1 (599 questions)

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Edexcel M1 Q1
  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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Edexcel M1 Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c762bd90-5b57-428a-a7a8-291a1a643a14-2_286_933_203_452} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a non-uniform beam \(A B\) of mass 10 kg and length 6 m resting in a horizontal position on a single support 2 m from \(A\). The beam is supported at \(B\) by a vertical string. Given that the magnitude of the tension in the string is 1.5 times the magnitude of the reaction at the support, find the distance of the centre of mass of the beam from \(A\).
(6 marks)
Edexcel M1 Q2
2. A ball of mass 2 kg moves on a smooth horizontal surface under the action of a constant force, \(\mathbf { F }\). The initial velocity of the ball is \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and 4 seconds later it has velocity \(( 10 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular, horizontal unit vectors.
  1. Making reference to the mass of the ball and the force it experiences, explain why it is reasonable to assume that the acceleration is constant.
  2. Find, giving your answers correct to 3 significant figures,
    1. the magnitude of the acceleration experienced by the ball,
    2. the angle which \(\mathbf { F }\) makes with the vector \(\mathbf { i }\).
Edexcel M1 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c762bd90-5b57-428a-a7a8-291a1a643a14-3_309_590_196_518} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a ball of mass 3 kg lying on a smooth plane inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 3 } { 5 }\). The ball is held in equilibrium by a force of magnitude \(P\) newtons, which acts at an angle of \(10 ^ { \circ }\) to the line of greatest slope of the plane.
  1. Suggest a suitable model for the ball. Giving your answers correct to 1 decimal place,
  2. find the value of \(P\),
  3. find the magnitude of the reaction between the ball and the plane.
Edexcel M1 Q4
4. A bullet of mass 50 g is fired horizontally at a wooden block of mass 4.95 kg which is lying at rest on a rough horizontal surface. The bullet enters the block at \(400 \mathrm {~ms} ^ { - 1 }\) and becomes embedded in the block.
  1. Find the speed with which the block begins to move. Given that the block decelerates uniformly to rest over a distance of 4 m ,
  2. show that the coefficient of friction is \(\frac { 2 } { g }\).
Edexcel M1 Q5
5. Two dogs, Fido and Growler, are playing in a field. Fido is moving in a straight line so that at time \(t\) his position vector relative to a fixed origin, \(O\), is given by \([ ( 2 t - 3 ) \mathbf { i } + t \mathbf { j } ]\) metres. Growler is stationary at the point with position vector \(( 2 \mathbf { i } + 5 \mathbf { j } )\) metres, where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.
  1. Find the displacement vector of Fido from Growler in terms of \(t\).
  2. Find the value of \(t\) for which the two dogs are closest.
  3. Find the minimum distance between the two dogs.
Edexcel M1 Q6
6. A particle moving in a straight line with speed \(5 U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) undergoes a uniform deceleration for 6 seconds which reduces its speed to \(2 \mathrm { Um } \mathrm { s } ^ { - 1 }\). It maintains this speed for 16 seconds before uniformly decelerating to rest in a further 2 seconds.
  1. Sketch a speed-time graph displaying this information.
  2. Find an expression for each of the decelerations in terms of \(U\). Given that the total distance travelled by the particle during this period of motion is 220 m ,
  3. find the value of \(U\).
Edexcel M1 Q7
7. A car of mass 1200 kg tows a trailer of mass 800 kg along a straight level road by means of a rigid towbar. The resistances to the motion of the car and the trailer are proportional to their masses. Given that the car experiences a resistance to motion of 300 N ,
  1. find the resistance to motion which the trailer experiences. Given that the engine of the car exerts a driving force of 3 kN ,
  2. find the acceleration of the system,
  3. show that the tension in the towbar is 1200 N . When the system has reached a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it continues at constant speed until an electrical fault causes the engine of the car to switch off. The brakes are used to apply a constant retarding force until the system comes to rest. Given that the retarding force of the brakes has magnitude 1 kN and assuming that the original resistances to motion of the car and the trailer remain constant,
  4. calculate the distance that the system travels during the braking period,
  5. find the magnitude and direction of the force exerted by the towbar on the car.
  6. Comment on the assumption that the original resistances to motion of the car and the trailer remain constant throughout the motion.
Edexcel M1 Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6fb27fe5-055a-4701-bd80-e66ebd57292a-2_403_550_214_609} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a light, inextensible string fixed at one end to a point \(P\). The other end is attached to a small object of weight 10 N . The object is subjected to a horizontal force \(H\) so that the string makes an angle of \(30 ^ { \circ }\) with the vertical.
  1. Find the magnitude of the tension in the string.
  2. Show that the ratio of the magnitude of the tension to the magnitude of \(H\) is \(2 : 1\).
Edexcel M1 Q2
2. A particle of mass 8 kg moves in a horizontal plane and is acted upon by three forces \(\mathbf { F } _ { 1 } = ( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 3 } = ( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.
  1. Find the magnitude, in newtons, of the resultant force which acts on the particle, giving your answer in the form \(k \sqrt { } 5\).
  2. Calculate, giving your answer in degrees correct to 1 decimal place, the angle the acceleration of the particle makes with the vector \(\mathbf { i }\).
Edexcel M1 Q3
3. A lorry accelerates uniformly from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 30 seconds.
  1. Find how far it travels while accelerating.
  2. Find, in seconds correct to 2 decimal places, the length of time it takes for the lorry to cover the first half of this distance.
    (6 marks)