A smooth sphere of mass m strikes a second sphere of mass 2m which is at rest. After the collision their direction of motion are at right angles. Then coefficient of restitution is-?

To determine the coefficient of restitution in this collision scenario, we need to analyze the situation using the principles of momentum and the definition of the coefficient of restitution itself. The coefficient of restitution (e) is a measure of how elastic a collision is, defined as the ratio of the relative speed of separation to the relative speed of approach along the line of impact.

Understanding the Collision

In this case, we have two spheres: one with mass m that is moving and strikes another sphere with mass 2m that is initially at rest. After the collision, they move at right angles to each other. This means that the momentum conservation principles will apply in both the x and y directions.

Setting Up the Problem

Let’s denote the initial velocity of the first sphere (mass m) as u and the final velocities of the two spheres after the collision as v1 for the first sphere and v2 for the second sphere. Since they move at right angles after the collision, we can assume the following:

• The first sphere moves in the x-direction.
• The second sphere moves in the y-direction.

Applying Conservation of Momentum

According to the law of conservation of momentum, the total momentum before the collision must equal the total momentum after the collision. Before the collision, the momentum is:

Initial Momentum = mu + 0 = mu

After the collision, the momentum can be expressed as:

Final Momentum = mv1 + 2mv2

Setting these equal gives us:

mu = mv1 + 2mv2

We can simplify this equation by dividing through by m:

u = v1 + 2v2

Using the Coefficient of Restitution

The coefficient of restitution is defined as:

e = (Relative speed of separation) / (Relative speed of approach)

In this case, the relative speed of approach before the collision is simply the speed of the first sphere, which is u. After the collision, since the spheres are moving at right angles, the relative speed of separation can be calculated using the Pythagorean theorem:

Relative speed of separation = sqrt(v1² + v2²)

Thus, we can express the coefficient of restitution as:

e = sqrt(v1² + v2²) / u

Finding the Velocities

From our momentum equation, we can express v1 in terms of v2:

v1 = u - 2v2

Substituting this into our equation for the coefficient of restitution:

e = sqrt((u - 2v2)² + v2²) / u

Expanding the square gives:

e = sqrt(u² - 4uv2 + 4v2² + v2²) / u

e = sqrt(u² - 4uv2 + 5v2²) / u

Solving for Coefficient of Restitution

To find a specific value for e, we need to consider the nature of the collision. In many cases, especially in elastic collisions, the coefficient of restitution can be calculated directly from the masses involved. For this problem, we can deduce that:

e = 1/2

This value indicates that the collision is partially elastic, as the two spheres do not separate with the same speed they approached each other.

In summary, the coefficient of restitution for this collision scenario is 1/2, reflecting the relationship between the velocities of the two spheres before and after the collision.