Question

A billiard ball of mass 0.40 kg rolls towards a cushion of a billiard table at +3.0 m/s and rebounds straight back at 3.0 m/s. If the collision force on the ball is -10 N, how long was the ball in contact with the cushion? Show all work.

a. 0.4 s
b. 0.2 s
c. 0.3 s
d. 0.1 s

Answer 1

A billiard ball of mass 0.40 kg rolls towards a cushion of a billiard table at +3.0 m/s and rebounds straight back at 3.0 m/s. If the collision force on the ball is -10 N, the ball will remain in contact with the cushion for 0.3 s (option C).

Step-by-step explanation:

To determine the time the ball was in contact with the cushion, we can use the impulse-momentum theorem, which states that the impulse (change in momentum) is equal to the force applied multiplied by the time of contact.

The impulse is given by the formula:

`\[ \text{Impulse} = \Delta p = m \cdot \Delta v \]`

where (m) is the mass of the ball and
`\(\Delta v\)` is the change in velocity.

Given that the mass of the ball is (m = 0.40kg) and the change in velocity is
`\(\Delta v = 3.0 \, \text{m/s} - (-3.0 \, \text{m/s}) = 6.0 \, \text{m/s}\),` we can calculate the impulse:

`\[ \text{Impulse} = 0.40 \, \text{kg} \cdot 6.0 \, \text{m/s} = 2.4 \, \text{Ns} \]`

Now, we can use the formula for impulse:

`\[ \text{Impulse} = F \cdot \Delta t \]`

Solving for
`\(\Delta t\)` (time of contact):

`\[ \Delta t = \frac{\text{Impulse}}{F} = \frac{2.4 \, \text{Ns}}{-10 \, \text{N}} = -0.24 \, \text{s} \]`

The negative sign indicates that the direction of the force is opposite to the direction of motion. Taking the absolute value, we get
`(0.24 \, \text{s}\),` which is approximately
`\(0.3 \, \text{s}\).` (option C).