A billiard ball rolling across a table to the right at 2.2 m/s makes a head-on elastic collision with an identical ball. The mass of a billiard ball is 32 g.

Question

If the second ball is initially at rest, what is the velocity of the first ball after the collision?
If the second ball is initially at rest, what is the velocity of the second ball after the collision?
If the second ball is initially moving to the left with a velocity of -1.1 m/s, what is the velocity of the first ball after the collision?
If the second ball is initially moving to the left with a velocity of -1.1 m/s, what is the velocity of the second ball after the collision?

Answer 1

1. If the second ball is initially at rest, the velocity of the first ball after the collision can be found using the formula for the final velocity of an object in an elastic collision:

v1f = ( (m1 - m2) / (m1 + m2) ) * v1i + ( (2 * m2) / (m1 + m2) ) * v2i

Since the two billiard balls have identical masses (m1 = m2) and the second ball is initially at rest (v2i = 0), the formula simplifies to:

v1f = ( (m1 - m2) / (m1 + m2) ) * v1i

v1f = ( (32 g - 32 g) / (32 g + 32 g) ) * 2.2 m/s

v1f = 0 m/s

So, the first ball will come to rest after the collision.

2. To find the velocity of the second ball after the collision, we use the same formula, but this time solving for v2f:

v2f = ( (2 * m1) / (m1 + m2) ) * v1i + ( (m2 - m1) / (m1 + m2) ) * v2i

Similarly, since m1 = m2 and v2i = 0, the formula simplifies to:

v2f = ( (2 * m1) / (m1 + m2) ) * v1i

v2f = ( (2 * 32 g) / (32 g + 32 g) ) * 2.2 m/s

v2f = 2.2 m/s

So, the second ball will have a velocity of 2.2 m/s to the right after the collision.

3. If the second ball is initially moving to the left with a velocity of -1.1 m/s, we can use the same formulas as before, but this time with v2i = -1.1 m/s:

v1f = ( (m1 - m2) / (m1 + m2) ) * v1i + ( (2 * m2) / (m1 + m2) ) * v2i

v1f = ( (32 g - 32 g) / (32 g + 32 g) ) * 2.2 m/s + ( (2 * 32 g) / (32 g + 32 g) ) * -1.1 m/s

v1f = 0 + (-1.1 m/s)

v1f = -1.1 m/s

So, the first ball will have a velocity of -1.1 m/s to the left after the collision.

4. To find the velocity of the second ball after the collision, we use the same formula as before but with v2i = -1.1 m/s:

v2f = ( (2 * m1) / (m1 + m2) ) * v1i + ( (m2 - m1) / (m1 + m2) ) * v2i

v2f = ( (2 * 32 g) / (32 g + 32 g) ) * 2.2 m/s + ( (32 g - 32 g) / (32 g + 32 g) ) * -1.1 m/s

v2f = 2.2 m/s + 0

v2f = 2.2 m/s

So, the second ball will have a velocity of 2.2 m/s to the right after the collision.