Projectile with plane collision

A question is this type if and only if it involves a particle projected under gravity that then strikes and rebounds from a plane surface, combining projectile motion with collision analysis.

4 questions · Standard +0.9

6.03k Newton's experimental law: direct impact
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Edexcel M4 Q3
8 marks Challenging +1.8
3. A smooth uniform sphere \(P\) of mass \(m\) is falling vertically and strikes a fixed smooth inclined plane with speed \(u\). The plane is inclined at an angle \(\theta , \theta < 45 ^ { \circ }\), to the horizontal. The coefficient of restitution between \(P\) and the inclined plane is \(e\). Immediately after \(P\) strikes the plane, \(P\) moves horizontally.
  1. Show that \(e = \tan ^ { 2 } \theta\).
Edexcel M4 2013 June Q2
6 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a3ae838-b58e-4957-8d98-f7d8a65df99a-03_604_741_123_605} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A smooth fixed plane is inclined at an angle \(\alpha\) to the horizontal. A smooth ball \(B\) falls vertically and hits the plane. Immediately before the impact the speed of \(B\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 1. Immediately after the impact the direction of motion of \(B\) is horizontal. The coefficient of restitution between \(B\) and the plane is \(\frac { 1 } { 3 }\). Find the size of angle \(\alpha\).
Edexcel M4 2017 June Q4
8 marks Standard +0.8
4. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] A line of greatest slope of a fixed smooth plane is parallel to the vector \(( - 4 \mathbf { i } - 3 \mathbf { j } )\). A particle \(P\) falls vertically and strikes the plane. Immediately before the impact, \(P\) has velocity \(- 7 \mathbf { j } \mathrm {~ms} ^ { - 1 }\). Immediately after the impact, \(P\) has velocity \(( - a \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(a\) is a positive constant.
  1. Show that \(a = 6\)
  2. Find the coefficient of restitution between \(P\) and the plane.
Edexcel M4 2005 June Q1
7 marks Standard +0.3
A small smooth ball of mass \(\frac{1}{2}\) kg is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{1}{3}\). Immediately before striking the plane the ball has speed 10 m s\(^{-1}\). The coefficient of restitution between ball and plane is \(\frac{1}{2}\). Find
  1. the speed, to 3 significant figures, of the ball immediately after the impact, [5]
  2. the magnitude of the impulse received by the ball as it strikes the plane. [2]