Particle-wall perpendicular collision

1. A smooth sphere is moving with speed \(U\) in a straight line on a smooth horizontal plane. It strikes a fixed smooth vertical wall at right angles. The coefficient of restitution between the sphere and the wall is \(\frac { 1 } { 2 }\).

Find the fraction of the kinetic energy of the sphere that is lost as a result of the impact.
(5 marks)

1 A small smooth sphere \(P\) of mass \(2 m\) is at rest on a smooth horizontal surface. A horizontal impulse of magnitude \(8 m u\) is given to \(P\). Subsequently \(P\) collides directly with a fixed smooth vertical barrier at right angles to \(P\) 's direction of motion. Given that the coefficient of restitution between \(P\) and the barrier is 0.75 , find the speed of \(P\) after the collision.

1 A particle \(P\) of mass 2.5 kg is moving with a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal plane when it collides directly with a fixed vertical wall. After the collision \(P\) moves away from the wall with a speed of \(2.8 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the coefficient of restitution between \(P\) and the wall.
2. Find the magnitude and state the direction of the impulse exerted on \(P\) by the wall.
3. State the magnitude and direction of the impulse exerted on the wall by \(P\).

2 A particle \(P\) of mass 4.5 kg is moving in a straight line on a smooth horizontal surface at a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\) when it strikes a vertical wall directly. It rebounds at a speed of \(1.6 \mathrm {~ms} ^ { - 1 }\).

1. Find the coefficient of restitution between \(P\) and the wall.
2. Determine the impulse applied to \(P\) by the wall, stating its direction.
3. Find the loss of kinetic energy of \(P\) as a result of the collision.
4. State, with a reason, whether the collision is perfectly elastic.

\begin{enumerate} \item A heavy ball, of mass 2 kg , rolls along a horizontal surface. It strikes a vertical wall at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) and rebounds. The coefficient of restitution between the ball and the wall is \(0 \cdot 4\). Find the kinetic energy lost in the impact. \item The velocity, \(v \mathrm {~ms} ^ { - 1 }\), of a particle at time \(t \mathrm {~s}\) is given by \(v = 4 t ^ { 2 } - 9\).

1. Find the acceleration of the particle when it is instantaneously at rest.
2. Find the distance travelled by the particle from time \(t = 0\) until it comes to rest. \item A particle \(P\) moves in a plane such that its position vector \(\mathbf { r }\) metres at time \(t\) seconds, relative to a fixed origin \(O\), is \(\mathbf { r } = \mathrm { e } ^ { t } \mathbf { i } - 2 t \mathbf { j }\).

1 A particle, of mass 0.8 kg , moves along a smooth horizontal surface. It hits a vertical wall, which is at right angles to the direction of motion of the particle, and rebounds. The speed of the particle as it hits the wall is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of restitution between the particle and the wall is 0.3 . Find

1. the impulse that the wall exerts on the particle,
2. the kinetic energy lost in the impact.

3 A disc of mass 0.5 kg is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal surface when it receives a horizontal impulse in a direction perpendicular to its direction of motion. Immediately after the impulse, the disc has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the impulse received by the disc.
2. Before the impulse, the disc is moving parallel to a smooth vertical wall, as shown in the diagram. \section*{11/1/1/1/1/1/1/1/1/1/1/1/ Wall} $$\overbrace { 3 \mathrm {~ms} ^ { - 1 } } ^ { \underset { < } { \bigcirc } } \text { Disc }$$ After the impulse, the disc hits the wall and rebounds with speed \(3 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Find the coefficient of restitution between the disc and the wall.
   [0pt] [4 marks]

1. A small ball of mass 0.3 kg is released from rest from a point 3.6 m above horizontal ground. The ball falls freely under gravity, hits the ground and rebounds vertically upwards.

In the first impact with the ground, the ball receives an impulse of magnitude 4.2 Ns . The ball is modelled as a particle.

1. Find the speed of the ball immediately after it first hits the ground.
2. Find the kinetic energy lost by the ball as a result of the impact with the ground.

1 A particle \(P\) of mass 4.5 kg is moving in a straight line on a smooth horizontal surface at a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\) when it strikes a vertical wall directly. It rebounds at a speed of \(1.6 \mathrm {~ms} ^ { - 1 }\).

1. Find the coefficient of restitution between \(P\) and the wall.
2. Determine the impulse applied to \(P\) by the wall, stating its direction.
3. Find the loss of kinetic energy of \(P\) as a result of the collision.
4. State, with a reason, whether the collision is perfectly elastic.