Two smooth spheres A and B of equal radii but of masses m and M respectively are free to move on a horizontal table. A is projected with speed u towards B which is at rest. On impact, the line joining their centres is inclined at an angle P to the velocity of A before impact. If e is the coefficient of restitution between the spheres, find the speed with which B begins to move?

To solve this problem, we need to analyze the collision between the two spheres, A and B, using the principles of momentum and the coefficient of restitution. Let's break this down step by step.

Understanding the Collision Dynamics

When sphere A, with mass m, collides with sphere B, which has mass M and is initially at rest, we can apply the laws of conservation of momentum and the definition of the coefficient of restitution (e). The angle P indicates the direction of the impact relative to the velocity of A.

Setting Up the Problem

Before the collision, sphere A has a velocity vector that we can denote as u. The velocity of sphere B is 0 since it is at rest. The angle P tells us how the line connecting the centers of the spheres is oriented with respect to the direction of A's velocity.

Components of Velocity

We can break down the velocity of A into two components:

• Perpendicular to the line of impact: u⊥ = u * sin(P)
• Along the line of impact: u∥ = u * cos(P)

Applying Conservation of Momentum

According to the conservation of momentum, the total momentum before the collision must equal the total momentum after the collision. The momentum along the line of impact can be expressed as:

m * u∥ + M * 0 = m * vA∥ + M * vB∥

Here, vA∥ is the velocity of sphere A after the collision along the line of impact, and vB∥ is the velocity of sphere B after the collision along the same line.

Using the Coefficient of Restitution

The coefficient of restitution (e) relates the velocities of the two spheres after the collision to their velocities before the collision:

e = (vB∥ - vA∥) / u∥

Solving the Equations

Now we have two equations:

1. From momentum conservation: m * u * cos(P) = m * vA∥ + M * vB∥
2. From the coefficient of restitution: vB∥ - vA∥ = e * u * cos(P)

We can rearrange the second equation to express vB∥ in terms of vA∥:

vB∥ = vA∥ + e * u * cos(P)

Substituting this into the momentum equation gives:

m * u * cos(P) = m * vA∥ + M * (vA∥ + e * u * cos(P))

Expanding this and simplifying leads to:

m * u * cos(P) = (m + M) * vA∥ + M * e * u * cos(P)

Rearranging terms, we can isolate vA∥:

vA∥ = (m * u * cos(P) - M * e * u * cos(P)) / (m + M)

Finding the Speed of Sphere B

Now, substituting vA∥ back into the equation for vB∥:

vB∥ = (m * u * cos(P) - M * e * u * cos(P)) / (m + M) + e * u * cos(P)

After simplifying, we find:

vB∥ = (m * u * cos(P) + M * (1 + e) * u * cos(P)) / (m + M)

This gives us the speed with which sphere B begins to move after the collision, taking into account both the momentum conservation and the coefficient of restitution.

Final Expression

Thus, the speed of sphere B after the collision can be expressed as:

vB∥ = (mu + Me) * u * cos(P) / (m + M)

This formula effectively captures the dynamics of the collision, allowing us to predict the motion of sphere B based on the initial conditions and the properties of the spheres involved.