Edexcel FM1 (Further Mechanics 1) 2023 June

1. A particle \(P\) of mass 2 kg is moving with velocity \(( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( - 6 \mathbf { i } + 42 \mathbf { j } )\) N s.
   1. Find the speed of \(P\) immediately after receiving the impulse.
   The angle through which the direction of motion of \(P\) has been deflected by the impulse is \(\alpha ^ { \circ }\)
2. Find the value of \(\alpha\)

1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car is a constant force of magnitude 400 N.

The engine of the car is working at a constant rate of 16 kW .

1. Find the value of \(U\). The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N . The resistance to the motion of the trailer is a constant force of magnitude 300 N . The engine of the car is working at a constant rate of 16 kW .
   The tow rope is modelled as being light and inextensible.
   Using the model,
2. find the tension in the tow rope at the instant when the speed of the car is \(\frac { 20 } { 3 } \mathrm {~ms} ^ { - 1 }\)

1. A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal plane. It collides directly with a particle \(Q\) of mass \(m\) that is moving on the plane with speed \(2 u\) in the opposite direction to \(P\).
   The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(e > \frac { 4 } { 5 }\)
   1. Show that the speed of \(Q\) immediately after the collision is \(\frac { ( 4 + 10 e ) u } { 3 }\)
   After the collision \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
2. Find, in terms of \(\boldsymbol { e }\), the set of values of \(f\) for which there will be a second collision between \(P\) and \(Q\).

1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(4 m g\). One end of the elastic string is attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the elastic string.
   The particle \(P\) hangs freely in equilibrium at the point \(E\), which is vertically below \(O\)
   1. Find the length \(O E\).
   Particle \(P\) is now pulled vertically downwards to the point \(A\), where \(O A = 4 a\), and released from rest. The resistance to the motion of \(P\) is a constant force of magnitude \(\frac { 1 } { 4 } m g\).
2. Find, in terms of \(a\) and \(g\), the speed of \(P\) after it has moved a distance \(a\). Particle \(P\) is now held at \(O\) Particle \(P\) is released from rest and reaches its maximum speed at the point \(B\). The resistance to the motion of \(P\) is again a constant force of magnitude \(\frac { 1 } { 4 } m g\).
3. Find the distance \(O B\).

5. \begin{figure}[h] \end{figure} A smooth uniform sphere \(S\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(S\) collides obliquely with another uniform sphere of mass \(M\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision the direction of motion of \(S\) makes an angle \(\alpha\), where \(0 < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(S\) makes an angle \(\beta\) with the line joining the centres of the spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

1. Show that \(\tan \beta = \frac { ( m + M ) \tan \alpha } { ( m - e M ) }\) Given that \(m = e M\),
2. show that the directions of motion of the two spheres immediately after the collision are perpendicular.

1. A particle \(P\) of mass \(m\) is falling vertically when it strikes a fixed smooth inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(0 < \alpha \leqslant 45 ^ { \circ }\)

At the instant immediately before the impact, the speed of \(P\) is \(u\).
At the instant immediately after the impact, \(P\) is moving horizontally with speed \(v\).

1. Show that the magnitude of the impulse exerted on the plane by \(P\) is \(m u \sec \alpha\) The coefficient of restitution between \(P\) and the plane is \(e\), where \(e > 0\)
2. Show that \(v ^ { 2 } = u ^ { 2 } \left( \sin ^ { 2 } \alpha + e ^ { 2 } \cos ^ { 2 } \alpha \right)\)
3. Show that the kinetic energy lost by \(P\) in the impact is $$\frac { 1 } { 2 } m u ^ { 2 } \left( 1 - e ^ { 2 } \right) \cos ^ { 2 } \alpha$$
4. Hence find, in terms of \(m\), \(u\) and \(e\) only, the kinetic energy lost by \(P\) in the impact.

7. \begin{figure}[h] \end{figure} A small smooth snooker ball is projected from the corner \(A\) of a horizontal rectangular snooker table \(A B C D\). The ball is projected so it first hits the side \(D C\) at the point \(P\), then hits the side \(C B\) at the point \(Q\) and then returns to \(A\). Angle \(A P D = \alpha\), Angle \(Q P C = \beta\), Angle \(A Q B = \gamma\)
The ball moves along \(A P\) with speed \(U\), along \(P Q\) with speed \(V\) and along \(Q A\) with speed \(W\), as shown in Figure 2. The coefficient of restitution between the ball and side \(D C\) is \(e _ { 1 }\)
The coefficient of restitution between the ball and side \(C B\) is \(e _ { 2 }\)
The ball is modelled as a particle. \section*{Use the model to answer all parts of this question.}

1. Show that \(\tan \beta = e _ { 1 } \tan \alpha\)
2. Hence show that \(e _ { 1 } \tan \alpha = e _ { 2 } \cot \gamma\)
3. By considering (angle \(A P Q\) + angle \(A Q P\) ) or otherwise, show that it would be possible for the ball to return to \(A\) only if \(e _ { 2 } > e _ { 1 }\) If instead \(e _ { 1 } = e _ { 2 }\), the ball would not return to \(A\).
   Given that \(e _ { 1 } = e _ { 2 }\)
4. use the result from part (b) to describe the path of the ball after it hits \(C B\) at \(Q\), explaining your answer.