Elastic Collisions in 2D (Edexcel A Level Further Maths: Further Mechanics 1): Exam Questions

[In this question $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere A has mass 2m kg and another smooth uniform sphere B, with the same radius as A, has mass 3m kg.

The spheres are moving on a smooth horizontal plane when they collide obliquely.

Immediately before the collision the velocity of A is $\left(3\mathbf{i}+3\mathbf{j}\right){\mathrm{ms}}^{-1}$ and the velocity of B is $\left(-5\mathbf{i}+2\mathbf{j}\right){\mathrm{ms}}^{-1}$.

At the instant of collision, the line joining the centres of the spheres is parallel to $\mathbf{i}$.

The coefficient of restitution between the spheres is $\frac{1}{4}$.

Find the velocity of B immediately after the collision.

Find, to the nearest degree, the size of the angle through which the direction of motion of B is deflected as a result of the collision.

Figure 2

Figure 2 represents the plan view of part of a horizontal floor, where AB and BC are fixed vertical walls with AB perpendicular to BC.

A small ball is projected along the floor towards AB with speed 6 ms^–1 on a path that makes an angle $\alpha$ with AB, where $\tan \alpha =\frac{4}{3}$. The ball hits AB and then hits BC.

Immediately after hitting AB, the ball is moving at an angle $\beta$ to AB, where $\tan \beta =\frac{1}{3}$.

The coefficient of restitution between the ball and AB is e.

The coefficient of restitution between the ball and BC is $\frac{1}{2}$.

By modelling the ball as a particle and the floor and walls as being smooth,

show that the value $e=\frac{1}{4}$.

Find the speed of the ball immediately after it hits BC.

Suggest two ways in which the model could be refined to make it more realistic.

[In this question $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere A has mass 0.2 kg and another smooth uniform sphere B, with the same radius as A, has mass 0.4 kg.

The spheres are moving on a smooth horizontal surface when they collide obliquely.
Immediately before the collision, the velocity of A is $\left(3\mathbf{i}+2\mathbf{j}\right){\mathrm{ms}}^{-1}$ and the velocity of B is $\left(-4\mathbf{i}-\mathbf{j}\right){\mathrm{ms}}^{-1}$.

At the instant of collision, the line joining the centres of the spheres is parallel to $\mathbf{i}$.

The coefficient of restitution between the spheres is $\frac{3}{7}$.

Find the velocity of A immediately after the collision. 

Find the magnitude of the impulse received by A in the collision. 

Find, to the nearest degree, the size of the angle through which the direction of motion of A is deflected as a result of the collision. 

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in a horizontal plane.]

Figure 1

Figure 1 represents the plan view of part of a smooth horizontal floor, where $AB$ represents a fixed smooth vertical wall.

A small ball of mass 0.5 kg is moving on the floor when it strikes the wall.

Immediately before the impact the velocity of the ball is $(7\mathbf{i}+2\mathbf{j}){\mathrm{ms}}^{-1}$.

Immediately after the impact the velocity of the ball is $(\mathbf{i}+6\mathbf{j}){\mathrm{ms}}^{-1}$.

The coefficient of restitution between the ball and the wall is $e$.

Show that $AB$ is parallel to $(2\mathbf{i}+3\mathbf{j})$.

Find the value of $e$.

A smooth uniform sphere $P$ has mass 0.3 kg. Another smooth uniform sphere $Q$, with the same radius as$P$, has mass 0.2 kg.

The spheres are moving on a smooth horizontal surface when they collide obliquely.
Immediately before the collision the velocity of $P$ is $(4\mathbf{i}+2\mathbf{j}){\mathrm{ms}}^{-1}$ and the velocity of $Q$ is $(–3\mathbf{i}+\mathbf{j}){\mathrm{ms}}^{-1}$.

At the instant of collision, the line joining the centres of the spheres is parallel to $\mathbf{i}$.

The kinetic energy of $Q$ immediately after the collision is half the kinetic energy of $Q$ immediately before the collision.

Find

i) the velocity of $P$ immediately after the collision,

ii) the velocity of $Q$ immediately after the collision,

iii) the coefficient of restitution between $P$ and $Q$,  

carefully justifying your answers.

Find the size of the angle through which the direction of motion of $P$ is deflected by the collision.

Figure 2

Figure 2 represents the plan view of part of a horizontal floor, where $AB$ and $CD$ represent fixed vertical walls, with $AB$parallel to $CD$.

A small ball is projected along the floor towards wall $AB$. Immediately before hitting wall $AB$, the ball is moving with speed $v$ ms^–1 at an angle $\alpha$ to $AB$, where $0<\alpha <\frac{\pi }{2}$.

The ball hits wall $AB$and then hits wall $CD$.

After the impact with wall $CD$, the ball is moving at angle $\frac{1}{2}\alpha$ to$CD$.

The coefficient of restitution between the ball and wall $AB$is $\frac{2}{3}$.

The coefficient of restitution between the ball and wall $CD$ is also $\frac{2}{3}$.

The floor and the walls are modelled as being smooth. The ball is modelled as a particle. 

Show that $\tan \left(\frac{1}{2}\alpha \right)=\frac{1}{3}$.

Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere $P$ has mass 0.3 kg. Another smooth uniform sphere $Q$, with the same radius as $P$, has mass 0.5 kg.

The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of $P$ is $(u\mathbf{i}+2\mathbf{j}){\mathrm{ms}}^{-1}$, where $u$ is a positive constant, and the velocity of $Q$ is $\left(-4\mathbf{i}+3\mathbf{j}\right){\mathrm{ms}}^{-1}$.

At the instant when the spheres collide, the line joining their centres is parallel to $\mathbf{i}$.

The coefficient of restitution between $P$ and $Q$ is $\frac{3}{5}$.

As a result of the collision, the direction of motion of $P$ is deflected through an angle of 90° and the direction of motion of $Q$ is deflected through an angle of $\alpha °$

Find the value of $u$.

Find the value of $\alpha$.

State how you have used the fact that $P$ and $Q$ have equal radii.

Figure 1

Figure 1 represents the plan view of part of a horizontal floor, where $AB$ and $BC$ represent fixed vertical walls, with $AB$ perpendicular to $BC$.

A small ball is projected along the floor towards the wall $AB$. Immediately before hitting the wall $AB$ the ball is moving with speed $v{\mathrm{ms}}^{-1}$ at an angle $\theta$ to $AB$.

The ball hits the wall $AB$ and then hits the wall $BC$. The coefficient of restitution between the ball and the wall $AB$ is $\frac{1}{3}$.

The coefficient of restitution between the ball and the wall $BC$ is $e$.

The floor and the walls are modelled as being smooth.

The ball is modelled as a particle.

The ball loses half of its kinetic energy in the impact with the wall $AB$. 

Find the exact value of $\cos \theta$.

The ball loses half of its remaining kinetic energy in the impact with the wall $BC$.

Find the exact value of $e$.

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors in a horizontal plane.]

Figure 3

Figure 3 represents the plan view of part of a smooth horizontal floor, where $AB$ is a fixed smooth vertical wall. 

The direction of $\overrightarrow{AB}$ is in the direction of the vector $(\mathbf{i}+\mathbf{j})$.

A small ball of mass 0.25 kg is moving on the floor when it strikes the wall $AB$.

Immediately before its impact with the wall $AB$, the velocity of the ball is $(8\mathbf{i}+2\mathbf{j}){\mathrm{ms}}^{-1}$.

Immediately after its impact with the wall $AB$, the velocity of the ball is $\mathbf{v}{\mathrm{ms}}^{-1}$.

The coefficient of restitution between the ball and the wall is  $\frac{1}{3}$.

By modelling the ball as a particle,

show that $\mathbf{v}=\left(4\mathbf{i}+6\mathbf{j}\right)$.

Find the magnitude of the impulse received by the ball in the impact.

Two smooth uniform spheres, $A$ and $B$, have equal radii. The mass of $A$ is $3m$and the mass of $B$ is $4m$. The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, $A$ is moving with speed $3u$ at 30° to the line of centres of the spheres and $B$ is moving with speed $2u$ at 30° to the line of centres of the spheres. The direction of motion of $B$ is turned through an angle of 90° by the collision, as shown in Figure 3.

(i) Find the size of the angle through which the direction of motion of $A$ is turned as a result of the collision.

(ii) Find, in terms of $m$ and $u$, the magnitude of the impulse received by $B$ in the collision.

Figure 5 represents the plan view of part of a smooth horizontal floor, where $RS$ and $ST$ are smooth fixed vertical walls. The vector $\overrightarrow{RS}$ is in the direction of $\mathbf{i}$ and the vector $\overrightarrow{ST}$ is in the direction of $(2\mathbf{i}+\mathbf{j})$.

A small ball $B$ is projected across the floor towards $RS$. Immediately before the impact with $RS$, the velocity of $B$ is $(6\mathbf{i}-8\mathbf{j}) \mathrm{m} {\mathrm{s}}^{-1}$. The ball bounces off $RS$ and then hits $ST$.

The ball is modelled as a particle.

Given that the coefficient of restitution between $B$ and $RS$is $e$, find the full range of possible values of $e$.

It is now given that $e=\frac{1}{4}$ and that the coefficient of restitution between $B$ and $ST$ is $\frac{1}{2}$.

Find, in terms of $\mathbf{i}$ and $\mathbf{j}$, the velocity of $B$ immediately after its impact with $ST$.

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 $0<\alpha <90°$, 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 $e$.

Show that $\tan \beta =\frac{(m+M)\tan \alpha }{(m-eM)}$ .

Given that $m=eM$, show that the directions of motion of the two spheres immediately after the collision are perpendicular.

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 $0<\alpha ⩽45°$

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 $mu\sec \alpha$

The coefficient of restitution between $P$ and the plane is $e$, where $e>0$

Show that $v^2=u^2({\sin }^2\alpha +e^2{\cos }^2\alpha )$

Show that the kinetic energy lost by $P$ in the impact is

$\frac{1}{2}m{u}^2(1-e^2){\cos }^2\alpha$

Hence find, in terms of $m$, $u$ and $e$ only, the kinetic energy lost by $P$ in the impact.

A small smooth snooker ball is projected from the corner $A$ of a horizontal rectangular snooker table $ABCD$.

The ball is projected so it first hits the side $DC$ at the point $P$, then hits the side $CB$ at the point $Q$ and then returns to $A$.

Angle $APD=\alpha$, Angle $QPC=\beta$, Angle $AQB=\gamma$.
The ball moves along $AP$ with speed $U$, along $PQ$ with speed $V$ and along $QA$ with speed $W$, as shown in Figure 2.

The coefficient of restitution between the ball and side $DC$ is $e_1$.

The coefficient of restitution between the ball and side $CB$ is $e_2$.

The ball is modelled as a particle.

Use the model to answer all parts of this question.

Show that $\tan \beta =e_1\tan \alpha$

Hence show that $e_1\tan \alpha =e_2\cot \gamma$

By considering (angle $APQ$ + angle $AQP$) 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 $e_1=e_2$, use the result from part (b) to describe the path of the ball after it hits $CB$ at $Q$, explaining your answer.

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal perpendicular unit vectors.]

A particle $P$ is moving with velocity $(4\mathbf{i}-\mathbf{j}) \mathrm{m} {\mathrm{s}}^{-1}$ on a smooth horizontal plane.
The particle collides with a smooth vertical wall and rebounds with velocity$(\mathbf{i}+3\mathbf{j}) \mathrm{m} {\mathrm{s}}^{-1}$

The coefficient of restitution between $P$ and the wall is $e$.

Find the value of $e$.

After the collision, $P$ goes on to hit a second smooth vertical wall, which is parallel to $\mathbf{i}$.

The coefficient of restitution between $P$ and this second wall is $\frac{1}{3}$

The angle through which the direction of motion of $P$ has been deflected by its collision with this second wall is $\alpha °$.

Find the value of $\alpha$, giving your answer to the nearest whole number.

A smooth uniform sphere $A$ of mass $m$ is moving with speed $U$ on a smooth horizontal plane. The sphere $A$ collides obliquely with a smooth uniform sphere $B$ of mass 3$m$ which is at rest on the plane. The two spheres have the same radius.

Immediately before the collision, the direction of motion of $A$makes an angle $\alpha$, where $0°<\alpha <90°$, with the line joining the centres of the spheres.

Immediately after the collision, the direction of motion of $A$ is perpendicular to its original direction, as shown in Figure 1.

The coefficient of restitution between the spheres is $e$.

Show that the speed of $B$ immediately after the collision is

$\frac{1}{4}(1+e)U\cos \alpha$

Show that $e>\frac{1}{3}$

Show that $0<\tan \alpha ⩽\frac{1}{\sqrt{2}}$

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal perpendicular unit vectors.]

A particle P is moving with velocity $(3\mathbf{i}-4\mathbf{j}){\text{ m s}}^{-1}$ on a smooth horizontal plane. The particle collides with a smooth vertical wall and rebounds with velocity $(-3\mathbf{i}+2\mathbf{j}){\text{ m s}}^{-1}$.

The coefficient of restitution between P and the wall is e.

Find the value of e.

After the collision, P goes on to hit a second smooth vertical wall, which is parallel to $\mathbf{j}$. The coefficient of restitution between P and this second wall is $\frac{1}{3}$.

The angle through which the direction of motion of P has been deflected by its collision with this second wall is $\alpha °$.

Find the value of $\alpha$, giving your answer to the nearest whole number.

A smooth uniform sphere A of mass $m$ is moving with speed U on a smooth horizontal plane. The sphere A collides obliquely with a smooth uniform sphere B of mass $2m$ which is at rest on the plane. The two spheres have the same radius.

Immediately before the collision, the direction of motion of A makes an angle $\alpha$, where $0°<\alpha <90°$, with the line joining the centres of the spheres. Immediately after the collision, the direction of motion of A is perpendicular to its original direction, as shown in Figure 1. The coefficient of restitution between the spheres is e.

Show that the speed of B immediately after the collision is $\frac{1}{3}(1+e)U\text{cos} \alpha$.

Show that $e>\frac{1}{2}$.

Show that $0<\text{tan} \alpha \le \frac{1}{\sqrt{3}}$.