Direct collision with direction reversal - Momentum and Collisions 1

OCR MEI M2 2008 January Q1

19 marks Moderate -0.3

A battering-ram consists of a wooden beam fixed to a trolley. The battering-ram runs along horizontal ground and collides directly with a vertical wall, as shown in Fig. 1.1. The battering-ram has a mass of 4000 kg. \includegraphics{figure_1} Initially the battering-ram is at rest. Some men push it for 8 seconds and let go just as it is about to hit the wall. While the battering-ram is being pushed, the constant overall force on it in the direction of its motion is 1500 N.
1. At what speed does the battering-ram hit the wall? [3]
The battering-ram hits a loose stone block of mass 500 kg in the wall. Linear momentum is conserved and the coefficient of restitution in the impact is 0.2.
1. Calculate the speeds of the stone block and of the battering-ram immediately after the impact. [6]
2. Calculate the energy lost in the impact. [3]
Small objects A and B are sliding on smooth, horizontal ice. Object A has mass 4 kg and speed 18 m s$^{-1}$ in the $\mathbf{i}$ direction. B has mass 8 kg and speed 9 m s$^{-1}$ in the direction shown in Fig. 1.2, where $\mathbf{i}$ and $\mathbf{j}$ are the standard unit vectors. \includegraphics{figure_2}
1. Write down the linear momentum of A and show that the linear momentum of B is $(36\mathbf{i} + 36\sqrt{3}\mathbf{j})$ N s. [2]
After the objects meet they stick together (coalesce) and move with a common velocity of $(u\mathbf{i} + v\mathbf{j})$ m s$^{-1}$.
1. Calculate $u$ and $v$. [3]
2. Find the angle between the direction of motion of the combined object and the $\mathbf{i}$ direction. Make your method clear. [2]

AQA Further AS Paper 2 Mechanics 2019 June Q7

12 marks Standard +0.3

Two smooth spheres, $P$ and $Q$, of equal radius are free to move on a smooth horizontal surface. The masses of $P$ and $Q$ are $3m$ and $m$ respectively. $P$ is set in motion with speed $u$ directly towards $Q$, which is initially at rest. $P$ subsequently collides with $Q$. \includegraphics{figure_7} Immediately after the collision, $P$ moves with speed $v$ and $Q$ moves with speed $w$. The coefficient of restitution between the spheres is $e$.

1. Show that $$v = \frac{u(3-e)}{4}$$ [4 marks]
2. Find $w$, in terms of $e$ and $u$, simplifying your answer. [2 marks]
Deduce that $$\frac{u}{2} \leq v \leq \frac{3u}{4}$$ [2 marks]
1. Find, in terms of $m$ and $u$, the maximum magnitude of the impulse that $P$ exerts on $Q$. [3 marks]
2. Describe the impulse that $Q$ exerts on $P$. [1 mark]

OCR MEI Further Mechanics Major 2019 June Q6

7 marks Challenging +1.2

\includegraphics{figure_6} The rim of a smooth hemispherical bowl is a circle of centre O and radius $a$. The bowl is fixed with its rim horizontal and uppermost. A particle P of mass $m$ is released from rest at a point A on the rim as shown in Fig. 6. When P reaches the lowest point of the bowl it collides directly with a stationary particle Q of mass $\frac{1}{2}m$. After the collision Q just reaches the rim of the bowl. Find the coefficient of restitution between P and Q. [7]

OCR MEI Further Mechanics Major 2024 June Q10

10 marks Challenging +1.2

A particle P of mass 2 kg is projected vertically upwards from horizontal ground with an initial speed of $14 \text{ m s}^{-1}$. At the same instant a particle Q of mass 8 kg is released from rest 5 m vertically above P. During the subsequent motion P and Q collide. The coefficient of restitution between P and Q is $\frac{11}{14}$. Determine the time between this collision and P subsequently hitting the ground. [10]

Pre-U Pre-U 9794/2 2011 June Q13

12 marks Challenging +1.2

\includegraphics{figure_13} Particles $A$ and $B$ of masses $2m$ and $m$, respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley $P$. The particle $A$ rests in equilibrium on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\alpha \leqslant 45°$ and $B$ is above the plane. The vertical plane defined by $APB$ contains a line of greatest slope of the plane, and $PA$ is inclined at angle $2\alpha$ to the horizontal (see diagram).

Show that the normal reaction $R$ between $A$ and the plane is $mg(2 \cos \alpha - \sin \alpha)$. [3]
Show that $R \geqslant \frac{1}{2}mg\sqrt{2}$. [3]

The coefficient of friction between $A$ and the plane is $\mu$. The particle $A$ is about to slip down the plane.

Show that $0.5 < \tan \alpha \leqslant 1$. [3]
Express $\mu$ as a function of $\tan \alpha$ and deduce its maximum value as $\alpha$ varies. [3]

Pre-U Pre-U 9794/3 2016 June Q10

7 marks Standard +0.3

A particle $A$ of mass $m$ travelling with speed $u$ on a smooth horizontal surface collides directly with a particle $B$ of mass $3m$ travelling with speed $\frac{2u}{5}$ in the opposite direction. After the collision, $A$ travels at speed $\frac{2u}{5}$ and $B$ travels at speed $\frac{4u}{15}$, both in the same direction as $B$ before the collision. Find $A$ and the coefficient of restitution between the two particles. [4]
A particle of mass 3 kg moving with velocity $(2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}) \text{ m s}^{-1}$ receives an impulse of $(6\mathbf{i} - 6\mathbf{j} - 9\mathbf{k})$ N s. Find the velocity of the particle after the impulse. [3]