Impulse on inclined plane - Momentum and Collisions 2

OCR M2 2013 January Q3

9 marks Standard +0.3

3 A particle $A$ is released from rest from the top of a smooth plane, which makes an angle of $30 ^ { \circ }$ with the horizontal. The particle $A$ collides 2 s later with a particle $B$, which is moving up a line of greatest slope of the plane. The coefficient of restitution between the particles is 0.4 and the speed of $B$ immediately before the collision is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 } . B$ has velocity $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ down the plane immediately after the collision. Find

the speed of $A$ immediately after the collision,
the distance $A$ moves up the plane after the collision. The masses of $A$ and $B$ are 0.5 kg and $m \mathrm {~kg}$, respectively.
Find the value of $m$.

OCR Further Mechanics AS 2018 June Q2

11 marks Moderate -0.3

2 A particle $P$ of mass 3.5 kg is moving down a line of greatest slope of a rough inclined plane. At the instant that its speed is $2.1 \mathrm {~ms} ^ { - 1 } P$ is at a point $A$ on the plane. At that instant an impulse of magnitude 33.6 Ns , directed up the line of greatest slope, acts on $P$.

Show that as a result of the impulse $P$ starts moving up the plane with a speed of $7.5 \mathrm {~ms} ^ { - 1 }$. While still moving up the plane, $P$ has speed $1.5 \mathrm {~ms} ^ { - 1 }$ at a point $B$ where $A B = 4.2 \mathrm {~m}$. The plane is inclined at an angle of $20 ^ { \circ }$ to the horizontal. The frictional force exerted by the plane on $P$ is modelled as constant.
Calculate the work done against friction as $P$ moves from $A$ to $B$.
Hence find the magnitude of the frictional force acting on $P$.
$P$ first comes to instantaneous rest at point $C$ on the plane.
Calculate $A C$.

OCR M3 2012 June Q2

8 marks Challenging +1.2

2
\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-3_442_636_255_715}
$B$ is a point on a smooth plane surface inclined at an angle of $15 ^ { \circ }$ to the horizontal. A particle $P$ of mass 0.45 kg is released from rest at the point $A$ which is 2.5 m vertically above $B$. The particle $P$ rebounds from the surface at an angle of $60 ^ { \circ }$ to the line of greatest slope through $B$, with a speed of $u \mathrm {~ms} ^ { - 1 }$. The impulse exerted on $P$ by the surface has magnitude $I$ Ns and is in a direction making an angle of $\theta ^ { \circ }$ with the upward vertical through $B$ (see diagram).

Explain why $\theta = 15$.
Find the values of $u$ and $I$.

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.

Show that $e = \tan ^ { 2 } \theta$.

Edexcel M4 2005 June Q1

7 marks Standard +0.3

A small smooth ball of mass $\frac { 1 } { 2 } \mathrm {~kg}$ is falling vertically. The ball strikes a smooth plane which is inclined at an angle $\alpha$ to the horizontal, where tan $\alpha = \frac { 3 } { 4 }$. Immediately before striking the plane the ball has speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. 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,
2. the magnitude of the impulse received by the ball as it strikes the plane.
3. A cyclist $P$ is cycling due north at a constant speed of $20 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. At 12 noon another cyclist $Q$ is due west of $P$. The speed of $Q$ is constant at $10 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Find the course which $Q$ should set in order to pass as close to $P$ as possible, giving your answer as a bearing.
  (5)
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6895ccda-84b8-45a5-9524-e5bfc37a2fee-2_437_1232_1174_443}
\end{figure} A smooth sphere $P$ lies at rest on a smooth horizontal plane. A second identical sphere $Q$, moving on the plane, collides with the sphere $P$. Immediately before the collision the direction of motion of $Q$ makes an angle $\alpha$ with the line joining the centres of the spheres. Immediately after the collision the direction of motion of $Q$ makes an angle $\beta$ with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is $e$. Show that $( 1 - e ) \tan \beta = 2 \tan \alpha$.

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 2014 June Q1

11 marks Challenging +1.2

A small smooth ball of mass $m$ is falling vertically when it strikes a fixed smooth plane which is inclined to the horizontal at an angle $\alpha$, where $0 ^ { \circ } < \alpha < 45 ^ { \circ }$. Immediately before striking the plane the ball has speed $u$. Immediately after striking the plane the ball moves in a direction which makes an angle of $45 ^ { \circ }$ with the plane. The coefficient of restitution between the ball and the plane is $e$. Find, in terms of $m , u$ and $e$, the magnitude of the impulse of the plane on the ball.
A ship $A$ is travelling at a constant speed of $30 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ on a bearing of $050 ^ { \circ }$. Another ship $B$ is travelling at a constant speed of $v \mathrm {~km} \mathrm {~h} ^ { - 1 }$ and sets a course to intercept $A$. At 1400 hours $B$ is 20 km from $A$ and the bearing of $A$ from $B$ is $290 ^ { \circ }$.
1. Find the least possible value of $v$.
Given that $v = 32$,
find the time at which $B$ intercepts $A$.

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.

Show that $a = 6$
Find the coefficient of restitution between $P$ and the plane.

Edexcel FM1 2023 June Q6

12 marks Challenging +1.2

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 $v$.

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 $e > 0$
Show that $v ^ { 2 } = u ^ { 2 } \left( \sin ^ { 2 } \alpha + e ^ { 2 } \cos ^ { 2 } \alpha \right)$
Show that the kinetic energy lost by $P$ in the impact is $$\frac { 1 } { 2 } m u ^ { 2 } \left( 1 - e ^ { 2 } \right) \cos ^ { 2 } \alpha$$
Hence find, in terms of $m$, $u$ and $e$ only, the kinetic energy lost by $P$ in the impact.