Questions FM1 (51 questions)

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Edexcel FM1 2019 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a871044a-17c5-440d-8d8f-886939603dd4-02_307_889_244_589} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 represents the plan of part of a smooth horizontal floor, where \(W _ { 1 }\) and \(W _ { 2 }\) are two fixed parallel vertical walls. The walls are 3 metres apart. A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leqslant 3\), from \(W _ { 1 }\) At time \(t = 0\), the particle is projected from \(O\) towards \(W _ { 1 }\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\)
The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.
  1. Show that \(T = \frac { 45 - 5 d } { 4 u }\) The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary.
  2. Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer.
Edexcel FM1 2019 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a871044a-17c5-440d-8d8f-886939603dd4-06_524_638_255_717} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) are fixed vertical walls with \(A B\) perpendicular to \(B C\). A small ball is projected along the floor towards \(A B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) on a path that makes an angle \(\alpha\) with \(A B\), where \(\tan \alpha = \frac { 4 } { 3 }\). The ball hits \(A B\) and then hits \(B C\).
Immediately after hitting \(A B\), the ball is moving at an angle \(\beta\) to \(A B\), where \(\tan \beta = \frac { 1 } { 3 }\)
The coefficient of restitution between the ball and \(A B\) is \(e\).
The coefficient of restitution between the ball and \(B C\) is \(\frac { 1 } { 2 }\)
By modelling the ball as a particle and the floor and walls as being smooth,
  1. show that the value of \(e = \frac { 1 } { 4 }\)
  2. find the speed of the ball immediately after it hits \(B C\).
  3. Suggest two ways in which the model could be refined to make it more realistic.
Edexcel FM1 2019 June Q3
  1. A particle \(P\), of mass 0.5 kg , is moving with velocity ( \(4 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse I of magnitude 2.5 Ns.
As a result of the impulse, the direction of motion of \(P\) is deflected through an angle of \(45 ^ { \circ }\) Given that \(\mathbf { I } = ( \lambda \mathbf { i } + \mu \mathbf { j } )\) Ns, find all the possible pairs of values of \(\lambda\) and \(\mu\).
Edexcel FM1 2019 June Q4
  1. A car of mass 600 kg pulls a trailer of mass 150 kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude 200 N . At the instant when the speed of the car is \(v \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(( 200 + \lambda v ) \mathrm { N }\), where \(\lambda\) is a constant.
When the engine of the car is working at a constant rate of 15 kW , the car is moving at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Show that \(\lambda = 8\) Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\)
    The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 200 N at all times. At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \(( 200 + 8 v ) \mathrm { N }\). The engine of the car is again working at a constant rate of 15 kW .
    When \(v = 10\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke.
  2. Find the acceleration of the car immediately after the towbar breaks.
  3. Use the work-energy principle to find the value of \(d\).
Edexcel FM1 2019 June Q5
  1. A particle \(P\) of mass \(3 m\) and a particle \(Q\) of mass \(2 m\) are moving along the same straight line on a smooth horizontal plane. The particles are moving in opposite directions towards each other and collide directly.
Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2 u\).
Immediately after the collision \(P\) and \(Q\) are moving in opposite directions.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Find the range of possible values of \(e\), justifying your answer. Given that \(Q\) loses 75\% of its kinetic energy as a result of the collision,
  2. find the value of \(e\).
Edexcel FM1 2019 June Q6
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A smooth uniform sphere \(A\) has mass 0.2 kg and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass 0.4 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 4 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\)
The coefficient of restitution between the spheres is \(\frac { 3 } { 7 }\)
  1. Find the velocity of \(A\) immediately after the collision.
  2. Find the magnitude of the impulse received by \(A\) in the collision.
  3. Find, to the nearest degree, the size of the angle through which the direction of motion of \(A\) is deflected as a result of the collision.
Edexcel FM1 2019 June Q7
  1. A particle \(P\), of mass \(m\), is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity kmg.
The other end of the spring is attached to a fixed point \(O\) on a ceiling.
The point \(A\) is vertically below \(O\) such that \(O A = 3 a\)
The point \(B\) is vertically below \(O\) such that \(O B = \frac { 1 } { 2 } a\)
The particle is held at rest at \(A\), then released and first comes to instantaneous rest at the point \(B\).
  1. Show that \(k = \frac { 4 } { 3 }\)
  2. Find, in terms of \(g\), the acceleration of \(P\) immediately after it is released from rest at \(A\).
  3. Find, in terms of \(g\) and \(a\), the maximum speed attained by \(P\) as it moves from \(A\) to \(B\).
Edexcel FM1 2020 June Q1
  1. A particle \(P\) of mass 0.5 kg is moving with velocity ( \(4 \mathbf { i } + 3 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(\mathbf { J }\) Ns. Immediately after receiving the impulse, \(P\) is moving with velocity \(( - \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
    1. Find the magnitude of \(\mathbf { J }\).
    The angle between the direction of the impulse and the direction of motion of \(P\) immediately before receiving the impulse is \(\alpha ^ { \circ }\)
  2. Find the value of \(\alpha\)
Edexcel FM1 2020 June Q2
  1. A truck of mass 1200 kg is moving along a straight horizontal road.
At the instant when the speed of the truck is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the truck is modelled as a force of magnitude \(( 900 + 9 v ) \mathrm { N }\). The engine of the truck is working at a constant rate of 25 kW .
  1. Find the deceleration of the truck at the instant when \(v = 25\) Later on, the truck is moving up a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\) At the instant when the speed of the truck is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the truck from non-gravitational forces is modelled as a force of magnitude ( \(900 + 9 v\) ) N. When the engine of the truck is working at a constant rate of 25 kW the truck is moving up the road at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the value of \(V\).
Edexcel FM1 2020 June Q3
  1. Two particles, \(A\) and \(B\), have masses \(3 m\) and \(4 m\) respectively. The particles are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\).
The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that the direction of motion of each of the particles is unchanged by the collision.
    (8) After the collision with \(A\), particle \(B\) collides directly with a third particle, \(C\), of mass \(2 m\), which is at rest on the surface. The coefficient of restitution between \(B\) and \(C\) is also \(e\).
  2. Show that there will be a second collision between \(A\) and \(B\).
Edexcel FM1 2020 June Q4
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{361d263e-0ee1-47e9-8fc2-0f127f1c2d7e-12_588_633_301_724} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 represents the plan view of part of a smooth horizontal floor, where \(A B\) represents a fixed smooth vertical wall. A small ball of mass 0.5 kg is moving on the floor when it strikes the wall.
Immediately before the impact the velocity of the ball is \(( 7 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Immediately after the impact the velocity of the ball is \(( \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
The coefficient of restitution between the ball and the wall is \(e\).
  1. Show that \(A B\) is parallel to \(( 2 \mathbf { i } + 3 \mathbf { j } )\).
  2. Find the value of \(e\).
Edexcel FM1 2020 June Q5
  1. A smooth uniform sphere \(P\) has mass 0.3 kg . Another smooth uniform sphere \(Q\), with the same radius as \(P\), has mass 0.2 kg .
The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of \(P\) is \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\).
The kinetic energy of \(Q\) immediately after the collision is half the kinetic energy of \(Q\) immediately before the collision.
  1. Find
    1. the velocity of \(P\) immediately after the collision,
    2. the velocity of \(Q\) immediately after the collision,
    3. the coefficient of restitution between \(P\) and \(Q\),
      carefully justifying your answers.
  2. Find the size of the angle through which the direction of motion of \(P\) is deflected by the collision.
Edexcel FM1 2020 June Q6
  1. A light elastic string with natural length \(l\) and modulus of elasticity \(k m g\) has one end attached to a fixed point \(A\) on a rough inclined plane. The other end of the string is attached to a package of mass \(m\).
The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\)
The package is initially held at \(A\). The package is then projected with speed \(\sqrt { 6 g l }\) up a line of greatest slope of the plane and first comes to rest at the point \(B\), where \(A B = 31\).
The coefficient of friction between the package and the plane is \(\frac { 1 } { 4 }\)
By modelling the package as a particle,
  1. show that \(k = \frac { 15 } { 26 }\)
  2. find the acceleration of the package at the instant it starts to move back down the plane from the point \(B\).
Edexcel FM1 2020 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{361d263e-0ee1-47e9-8fc2-0f127f1c2d7e-24_553_951_258_557} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 represents the plan view of part of a horizontal floor, where \(A B\) and \(C D\) represent fixed vertical walls, with \(A B\) parallel to \(C D\). A small ball is projected along the floor towards wall \(A B\). Immediately before hitting wall \(A B\), the ball is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to \(A B\), where \(0 < \alpha < \frac { \pi } { 2 }\) The ball hits wall \(A B\) and then hits wall \(C D\).
After the impact with wall \(C D\), the ball is moving at angle \(\frac { 1 } { 2 } \alpha\) to \(C D\).
The coefficient of restitution between the ball and wall \(A B\) is \(\frac { 2 } { 3 }\)
The coefficient of restitution between the ball and wall \(C D\) is also \(\frac { 2 } { 3 }\)
The floor and the walls are modelled as being smooth. The ball is modelled as a particle.
  1. Show that \(\tan \left( \frac { 1 } { 2 } \alpha \right) = \frac { 1 } { 3 }\)
  2. Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.
Edexcel FM1 2021 June Q1
  1. A van of mass 900 kg is moving along a straight horizontal road.
At the instant when the speed of the van is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van is modelled as a force of magnitude \(( 500 + 7 v ) \mathrm { N }\). When the engine of the van is working at a constant rate of 18 kW , the van is moving along the road at a constant speed \(V \mathrm {~ms} ^ { - 1 }\)
  1. Find the value of \(V\). Later on, the van is moving up a straight road that is inclined to the horizontal at an angle \(\theta\), where \(\sin \theta = \frac { 1 } { 21 }\) At the instant when the speed of the van is \(v \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude \(( 500 + 7 v ) \mathrm { N }\). The engine of the van is again working at a constant rate of 18 kW .
  2. Find the acceleration of the van at the instant when \(v = 15\)
Edexcel FM1 2021 June Q2
  1. Two particles, \(A\) and \(B\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
Particle \(A\) has mass \(5 m\) and particle \(B\) has mass \(3 m\).
The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\)
Immediately after the collision the speed of \(A\) is \(v\) and the speed of \(B\) is \(2 v\).
Given that \(A\) and \(B\) are moving in the same direction after the collision,
  1. find the set of possible values of \(e\). Given also that the kinetic energy of \(A\) immediately after the collision is \(16 \%\) of the kinetic energy of \(A\) immediately before the collision,
  2. find
    1. the value of \(e\),
    2. the magnitude of the impulse received by \(A\) in the collision, giving your answer in terms of \(m\) and \(v\).
Edexcel FM1 2021 June Q3
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A smooth uniform sphere \(P\) has mass 0.3 kg . Another smooth uniform sphere \(Q\), with the same radius as \(P\), has mass 0.5 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of \(P\) is \(( u \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(u\) is a positive constant, and the velocity of \(Q\) is \(( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At the instant when the spheres collide, the line joining their centres is parallel to \(\mathbf { i }\).
The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 3 } { 5 }\)
As a result of the collision, the direction of motion of \(P\) is deflected through an angle of \(90 ^ { \circ }\) and the direction of motion of \(Q\) is deflected through an angle of \(\alpha ^ { \circ }\)
  1. Find the value of \(u\)
  2. Find the value of \(\alpha\)
  3. State how you have used the fact that \(P\) and \(Q\) have equal radii.
Edexcel FM1 2021 June Q4
  1. A particle \(P\) has mass 0.5 kg . It is moving in the \(x y\) plane with velocity \(8 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(\lambda ( - \mathbf { i } + \mathbf { j } )\) Ns, where \(\lambda\) is a positive constant.
The angle between the direction of motion of \(P\) immediately before receiving the impulse and the direction of motion of \(P\) immediately after receiving the impulse is \(\theta ^ { \circ }\) Immediately after receiving the impulse, \(P\) is moving with speed \(4 \sqrt { 10 } \mathrm {~ms} ^ { - 1 }\)
Find (i) the value of \(\lambda\)
(ii) the value of \(\theta\)
Edexcel FM1 2021 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7f077b82-6b39-4cb5-8574-bfa308c88df3-16_575_665_246_699} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) represent fixed vertical walls, with \(A B\) perpendicular to \(B C\). A small ball is projected along the floor towards the wall \(A B\). Immediately before hitting the wall \(A B\) the ball is moving with speed \(v \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) to \(A B\). The ball hits the wall \(A B\) and then hits the wall \(B C\).
The coefficient of restitution between the ball and the wall \(A B\) is \(\frac { 1 } { 3 }\)
The coefficient of restitution between the ball and the wall \(B C\) is \(e\).
The floor and the walls are modelled as being smooth.
The ball is modelled as a particle.
The ball loses half of its kinetic energy in the impact with the wall \(A B\).
  1. Find the exact value of \(\cos \theta\). The ball loses half of its remaining kinetic energy in the impact with the wall \(B C\).
  2. Find the exact value of \(e\).
Edexcel FM1 2021 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7f077b82-6b39-4cb5-8574-bfa308c88df3-20_401_814_246_628} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light elastic spring has natural length \(3 l\) and modulus of elasticity \(3 m g\).
One end of the spring is attached to a fixed point \(X\) on a rough inclined plane.
The other end of the spring is attached to a package \(P\) of mass \(m\).
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\)
The package is initially held at the point \(Y\) on the plane, where \(X Y = l\). The point \(Y\) is higher than \(X\) and \(X Y\) is a line of greatest slope of the plane, as shown in Figure 2. The package is released from rest at \(Y\) and moves up the plane.
The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 3 }\)
By modelling \(P\) as a particle,
  1. show that the acceleration of \(P\) at the instant when \(P\) is released from rest is \(\frac { 17 } { 15 } \mathrm {~g}\)
  2. find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the spring first reaches its natural length of 31 .
Edexcel FM1 2021 June Q7
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7f077b82-6b39-4cb5-8574-bfa308c88df3-24_543_789_294_639} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 represents the plan view of part of a smooth horizontal floor, where \(A B\) is a fixed smooth vertical wall. The direction of \(\overrightarrow { A B }\) is in the direction of the vector \(( \mathbf { i } + \mathbf { j } )\)
A small ball of mass 0.25 kg is moving on the floor when it strikes the wall \(A B\).
Immediately before its impact with the wall \(A B\), the velocity of the ball is \(( 8 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
Immediately after its impact with the wall \(A B\), the velocity of the ball is \(\mathbf { v m s } ^ { - 1 }\)
The coefficient of restitution between the ball and the wall is \(\frac { 1 } { 3 }\)
By modelling the ball as a particle,
  1. show that \(\mathbf { v } = 4 \mathbf { i } + 6 \mathbf { j }\)
  2. Find the magnitude of the impulse received by the ball in the impact.
Edexcel FM1 2022 June Q1
  1. A particle \(A\) of mass \(3 m\) and a particle \(B\) of mass \(m\) are moving along the same straight line on a smooth horizontal surface. The particles are moving in opposite directions towards each other when they collide directly.
Immediately before the collision, the speed of \(A\) is \(k u\) and the speed of \(B\) is \(u\). Immediately after the collision, the speed of \(A\) is \(v\) and the speed of \(B\) is \(2 v\). The magnitude of the impulse received by \(B\) in the collision is \(\frac { 3 } { 2 } m u\).
  1. Find \(v\) in terms of \(u\) only.
  2. Find the two possible values of \(k\).
Edexcel FM1 2022 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-06_287_846_246_612} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A van of mass 600 kg is moving up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\). The van is towing a trailer of mass 150 kg . The van is attached to the trailer by a towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 200 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 100 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 12 kW .
Find the tension in the towbar at the instant when the speed of the van is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Edexcel FM1 2022 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-10_302_442_244_813} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 0.5 kg is moving in a straight line with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude 3 Ns .
The angle between the direction of motion of \(P\) immediately before receiving the impulse and the line of action of the impulse is \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. Find the speed of \(P\) immediately after receiving the impulse.
Edexcel FM1 2022 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-12_387_929_246_568} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two smooth uniform spheres, \(A\) and \(B\), have equal radii. The mass of \(A\) is \(3 m\) and the mass of \(B\) is \(4 m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, \(A\) is moving with speed \(3 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres and \(B\) is moving with speed \(2 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(90 ^ { \circ }\) by the collision, as shown in Figure 3.
  1. Find the size of the angle through which the direction of motion of \(A\) is turned as a result of the collision.
  2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(B\) in the collision.