Question

A smooth uniform sphere is moving on a smooth horizontal surface when it collides with a smooth vertical wall. Before the impact the direction of motion of the sphere makes an angle of 60° with the wall. After the impact the direction of motion of the sphere makes an angle of 30° with the wall. Determine the value of the coefficient of restitution between the sphere and the wall.​

Answer 1

e = 1/√3.

Step-by-step explanation:

When the sphere hits the wall , the wall exerts a force normal to its surface. So there is no force exerted in the direction parallel to the surface of the wall. Hence we can apply the principle of conservation of linear momentum for the impact in the direction parallel to the surface of the wall. Let the mass of the sphere be m. Let the initial and final velocities of the sphere be u and v respectively.

So m * u Cos 60° = m * v Cos 30°

v/u = Cos 60° / Cos 30° = (1/2) / (√3/2) = 1/√3.

Coefficient of restitution is the ratio of final velocity after the impact to the velocity before the impact.

So answer = e = 1/√3

Answer 2

The coefficient of restitution (e) is 0.

This means that the collision between the smooth uniform sphere and the smooth vertical wall is completely elastic, with no loss of kinetic energy.

Step-by-step explanation:

To determine the coefficient of restitution between the sphere and the wall, let's use the conservation of momentum and energy principles. When the sphere collides with the wall, both momentum and kinetic energy are conserved in an elastic collision (no loss of kinetic energy).

Let's denote the initial velocity of the sphere before the impact as V_i and the final velocity after the impact as V_f. The angle between the initial velocity and the wall is 60°, and after the impact, it becomes 30°.

Step 1: Conservation of Momentum

Before the impact, the x-component of the momentum is V_i * cos(60°), and after the impact, it becomes V_f * cos(30°). Since there is no motion in the y-direction (perpendicular to the wall), the y-component of momentum remains the same.

Conservation of x-component momentum:

V_i * cos(60°) = V_f * cos(30°)

Step 2: Conservation of Kinetic Energy

As mentioned earlier, since it's an elastic collision, the kinetic energy before and after the impact remains the same.

Kinetic energy before the impact:

(1/2) * m * V_i^2

Kinetic energy after the impact:

(1/2) * m * V_f^2

where 'm' is the mass of the sphere.

Step 3: Relationship between Kinetic Energy and Momentum

For a uniform sphere, the kinetic energy can be related to momentum as follows:

Kinetic energy = (1/2) * m * V^2

Momentum = m * V

where 'V' is the magnitude of the velocity.

Using the relationship between kinetic energy and momentum, we can write:

(1/2) * m * V_i^2 = (1/2) * m * V_f^2

Step 4: Combine the Equations

We can now combine the equations to solve for the coefficient of restitution (e):

V_i * cos(60°) = V_f * cos(30°)

(1/2) * m * V_i^2 = (1/2) * m * V_f^2

Since the mass 'm' appears on both sides of the second equation, it cancels out.

Now, let's solve these two equations simultaneously:

V_i * cos(60°) = V_f * cos(30°)

V_i^2 = V_f^2

Since V_i^2 = V_f^2, we can take the square root of both sides:

V_i = V_f

Step 5: Calculate the Coefficient of Restitution

The coefficient of restitution (e) is defined as the ratio of relative velocities after and before the collision:

e = (relative velocity after collision) / (relative velocity before collision)

In this case, since the velocities are the same before and after the collision (V_i = V_f), the relative velocity is zero.

Therefore, the coefficient of restitution (e) is 0.

This means that the collision between the smooth uniform sphere and the smooth vertical wall is completely elastic, with no loss of kinetic energy.