A ball of mass 1000g moving with 7m/s impinges another ball of twice its own mass

and moving with 1/6th of its own velocity and moves in the opposite direction. If the coefficient of restitution is 0.75, calculate the velocities of the two balls after impact.
Plase solve this pronlem..

To solve this problem, we can use the principle of conservation of momentum and the coefficient of restitution.

The principle of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces act on the system.

Let's denote the mass of the first ball as m1, the mass of the second ball as m2, the initial velocity of the first ball as v1, and the initial velocity of the second ball as v2.

According to the problem, m1 = 1000g = 1kg, m2 = 2m1 = 2kg, v1 = 7 m/s, and v2 = v1/6 = 7/6 m/s.

Let's assume the final velocities of the first ball and the second ball are v1' and v2' respectively.

Now, we can use the conservation of momentum to write the equation:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')

Substituting the given values, we have:
(1 * 7) + (2 * 7/6) = (1 * v1') + (2 * v2')

Simplifying, we get:
7 + 7/3 = v1' + 2/3 * v2'

Now, let's use the coefficient of restitution. The coefficient of restitution (e) is the ratio of the relative velocity after the collision to the relative velocity before the collision. It is given as 0.75.

The relative velocity before the collision is v2 - v1 (since they're moving in opposite directions), and the relative velocity after the collision is v2' - v1'.

Using the coefficient of restitution, we can write the equation:
e = (v2' - v1') / (v2 - v1)

Substituting the given values, we have:
0.75 = (v2' - v1') / (v2 - v1)

Now, let's solve for v1' and v2' using the two equations we have.

From equation 1: 7 + 7/3 = v1' + 2/3 * v2'

Simplifying, we have:
21/3 + 7/3 = v1' + 2/3 * v2'
(28/3) = v1' + (2/3) * v2'

Now, let's isolate v1' in terms of v2' using equation 2.

From equation 2: 0.75 = (v2' - v1') / (v2 - v1)

Simplifying, we have:
0.75 * (v2 - v1) = (v2' - v1')
0.75 * (7/6 - 7) = (v2' - v1')

Multiply both sides by 6:
0.75 * (7 - 42/6) = 6 * (v2' - v1')
0.75 * (42/6 - 42/6) = (v2' - v1')
0.75 * 0 = (v2' - v1')
0 = (v2' - v1')

Now, substituting v2' - v1' = 0 into equation 1, we have:
28/3 = v1' + (2/3) * v2'

Since (v2' - v1') = 0, then v1' must be equal to v2'.

From this, we can conclude that the final velocities of both balls will be equal.

Therefore, the velocity of both balls after the impact will be 28/3 m/s in opposite directions.

Please note that if there are any unit conversions needed, make sure to perform them before calculating the final answer.