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A sphere of mass m moving with constant velocity u, collides with another stationary sphere of same mass. If e is the coefficient of restitution, the ratio of the final velocities of the first and second spheres is
  • a)
    1 + e/1 − e
  • b)
    1 − e/1 + e
  • c)
    e/1 − e
  • d)
    1 + e/e
Correct answer is option 'B'. Can you explain this answer?
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The coefficient of restitution, e, is defined as the ratio of the relative velocities of the spheres after the collision to the relative velocities before the collision.

Let V1 and V2 be the final velocities of the first and second spheres, respectively.

Since the first sphere is moving with constant velocity u, its initial relative velocity before the collision is u - 0 = u.

After the collision, the first sphere will still be moving with constant velocity u, so its final relative velocity will be u - 0 = u.

The second sphere is initially at rest, so its initial relative velocity before the collision is 0 - u = -u.

Using the definition of the coefficient of restitution, we have:

e = (V2 - V1) / (-u)

Rearranging the equation, we get:

V2 - V1 = -e * u

Dividing both sides by V1, we get:

(V2 - V1) / V1 = -e * u / V1

Since V1 is non-zero (as given in the problem), we can multiply both sides by (1 / V1):

(V2 - V1) / V1 = -e * u / V1 * (1 / V1)

Simplifying, we get:

(V2 - V1) / V1 = -e * u / (V1^2)

Multiplying both sides by V1, we get:

V2 - V1 = -e * u / (V1^2) * V1

Simplifying, we get:

V2 - V1 = -e * u / V1

Adding V1 to both sides, we get:

V2 = V1 - e * u / V1

So, the ratio of the final velocities of the first and second spheres is:

V1 / V2 = V1 / (V1 - e * u / V1)

Therefore, the ratio of the final velocities of the first and second spheres is 1 / (1 - e * u / V1).
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