A heavy Steel ball is struck by a small Steel pellet moving horizontally with velocity 20 m per second. If the pellet bounces of the Steel ball with no slippage and then rises vertically to a height 10 metre above the point of contact, then what is hundred times the elastic coefficient of restitution(e) i.e. 100e?

Question

Answer

Analysis:
To solve this problem, we need to consider the conservation of momentum and the conservation of mechanical energy.

Conservation of Momentum:
When the small steel pellet strikes the heavy steel ball, the momentum before and after the collision should be conserved. Since the pellet moves horizontally and the ball is initially at rest, the total momentum before the collision is given by:
Initial momentum = m1 * v1 = m1 * 20

After the collision, the pellet bounces off the ball with no slippage, so the direction of its velocity changes. Let's assume the velocity of the pellet after the collision is v2.

Final momentum = m1 * v1 + m2 * v2

Since the ball is initially at rest, its momentum is zero. Therefore, we have:
m1 * v1 = -m2 * v2

Conservation of Mechanical Energy:
The pellet then rises vertically to a height of 10 meters above the point of contact. At this point, the kinetic energy of the pellet is converted into potential energy. Since the pellet is at its maximum height, its velocity is zero, and all of its initial kinetic energy is converted into potential energy.

Initial kinetic energy = 0.5 * m2 * v2^2
Final potential energy = m2 * g * h

Where g is the acceleration due to gravity and h is the height.

Elastic Coefficient of Restitution:
The coefficient of restitution (e) is a measure of the elasticity of a collision. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach.
e = (v2f - v1f) / (v1i - v2i)

Since the pellet bounces off the ball with no slippage, the final velocity of the pellet in the horizontal direction (v2f) is equal to its initial velocity (v1i). Therefore, the coefficient of restitution can be simplified to:
e = (v1i - v2i) / (v1i - v2i) = 1

Therefore, 100e = 100 * 1 = 100.

Conclusion:
The hundred times the elastic coefficient of restitution is 100. This means that the collision between the small steel pellet and the heavy steel ball is perfectly elastic, resulting in a transfer of momentum and no loss of mechanical energy.

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