A ball with a mass of #7 kg# moving at #7 m/s# hits a still ball with a mass of #8 kg#. If the first ball stops moving, how fast is the second ball moving?

Question

A ball with a mass of #7 kg# moving at #7 m/s# hits a still ball with a mass of #8 kg#.  If the first ball stops moving, how fast is the second ball moving?

Leo Abeyta · Accepted Answer

The second ball is moving at #6.5# m/s.

Notation here uses #T_1# as the kinetic energy of the first ball and #T_2# as the kinetic energy of the second.

Procedure

Find the kinetic energy of the first ball using the formula below

i. As the first ball has completely stopped moving and the second ball was not moving to start with, the second ball must now have the exact amount of energy that the first had. This assumes the first ball transfers its energy perfectly and that no energy is lost to friction, sound generation, heat generation etc. This is a reasonable assumption to make at this scale.
ii. Substitute the kinetic energy that the first ball was found to have into the equation of kinetic energy for the second ball.

Solve for the second ball's velocity.

As only one significant figure was given for the numbers in the question, the final answer should actually be rounded to #7# m/s. I rounded to #6.5# m/s (by assuming that the masses and velocity were #7.0# kg, #8.0# kg and #7.0# m/s, i.e. known accurately) so that the difference in velocity between the two balls was actually noticeable. As the second ball is heavier, the same amount of kinetic energy will cause it to move more slowly.

Sebastian Acosta · Answer

undefined To find the velocity of the second ball after the collision, we can use the principle of conservation of momentum:

m1 * v1(initial) + m2 * v2(initial) = m1 * v1(final) + m2 * v2(final)

Given:
m1 = 7 kg (mass of the first ball)
v1(initial) = 7 m/s (initial velocity of the first ball)
m2 = 8 kg (mass of the second ball)
v1(final) = 0 m/s (final velocity of the first ball, since it stops)
v2(final) = unknown (final velocity of the second ball)

Plugging in the values:

7 kg * 7 m/s + 8 kg * 0 m/s = 7 kg * 0 m/s + 8 kg * v2(final)

Solving for v2(final):

49 kg m/s = 8 kg * v2(final)

v2(final) = 49 kg m/s / 8 kg
v2(final) = 6.125 m/s

So, the second ball is moving at approximately 6.125 m/s after the collision.