Oblique impact

Two uniform spheres P and Q have equal radii. The mass of P is 0.3 kg and the mass of Q is 0.4 kg. The spheres are moving on a smooth horizontal plane when they collide obliquely.

Immediately before the collision,

• P is moving with speed 5 ms¹ at 30° to the line of centres of the spheres

• Q is moving with speed 3 ms¹ at 60° to the line of centres of the spheres

Immediately after the collision,

• P is moving with speed v ms¹ at 60° to the line of centres of the spheres

• Q is moving with speed w ms¹ at θ to the line of centres of the spheres

as in the plan view shown in Figure 4.

Find (i) the value of w

(ii) the value of θ

Find the coefficient of restitution between P and Q.

Figure 5 represents the plan view of part of a smooth horizontal floor, where RS and ST are smooth fixed vertical walls. The vector RS→ is in the direction of the vector i. The vector ST→ is in the direction of the vector 3i 4j.

A small ball B of mass 0 25. kg is projected across the floor towards RS.

Immediately before the impact with RS, the velocity of B is v ms¹ Immediately after the impact with RS, the velocity of B is 3i 2j ms₁ The coefficient of restitution between B and RS is The ball is modelled as a particle.

Show that the kinetic energy lost by B in the impact with RS is 4 J.

Immediately after the impact with ST, the velocity of B is w ms¹ Given that the coefficient of restitution between B and ST is

find w in terms of i and j.

[In this question, i and j are horizontal perpendicular unit vectors.]

A particle P is moving with velocity (4i – j) m s^–1 on a smooth horizontal plane. The particle collides with a smooth vertical wall and rebounds with velocity (i + 3j) 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 i. The coefficient of restitution between P and this second wall is

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

Find the value of α, 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 α, where 0° < α < 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

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 α, where 0 < α < 90^o, with the line joining the centres of the spheres.

Immediately after the collision the direction of motion of S makes an angle β with the line joining the centres of the spheres, as shown in Figure 1.

The coefficient of restitution between the spheres is e.

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 α, where 0 < 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

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

Show that v² u²(sin² e² cos²)

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

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 = α, Angle QPC = β, Angle AQB = γ 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 e₁

The coefficient of restitution between the ball and side CB is e₂

The ball is modelled as a particle.

Use the model to answer all parts of this question.

Show that tan e₁ tan

Hence show that e₁ tan e₂ cot

By considering (angle APQ + angle AQP) or otherwise, show that it would be possible for the ball to return to A only if e₂ > e₁

If instead e₁=e₂, the ball would not return to A.

Given that e₁=e₂

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, i and 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 (ui + 2j)m s⁻¹, where u is a positive constant, and the velocity of Q is (−4i + 3j)m s⁻¹

At the instant when the spheres collide, the line joining their centres is parallel to i. The coefficient of restitution between P and Q is

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 α °

Find the value of u

Find the value of α

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

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 vm s⁻¹ at an angle θ 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

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 cosθ.

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, i and j are perpendicular unit vectors in a horizontal plane.]

Figure 3 represents the plan view of part of a smooth horizontal floor, where AB is a fixed smooth vertical wall. The direction of AB is in the direction of the vector (i + 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 (8i + 2j)m s⁻¹

Immediately after its impact with the wall AB, the velocity of the ball is vm s⁻¹ The coefficient of restitution between the ball and the wall is

By modelling the ball as a particle,

show that v = 4i + 6j

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

[In this question, i and j are perpendicular unit vectors in a horizontal plane.]

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 (7i + 2j) m s^–1.

Immediately after the impact the velocity of the ball is (i + 6j) m s^–1.

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

Show that AB is parallel to (2i + 3j).

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 (4i + 2j) m s^–1 and the velocity of Q is (–3i + j) m s^–1.

At the instant of collision, the line joining the centres of the spheres is parallel to i.

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

Find

the velocity of P immediately after the collision,

the velocity of Q immediately after the collision,

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 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 vm s^–1 at an angle α to AB, where 0 < α < The ball hits wall AB and then hits wall CD. After the impact with wall CD, the ball is moving at angle α to CD. The coefficient of restitution between the ball and wall AB is The coefficient of restitution between the ball and wall CD is also The floor and the walls are modelled as being smooth. The ball is modelled as a particle.

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