91Ó°ÊÓ

Impulse is an important concept in physics, especially when you want to understand how forces affect the motion of objects. In simple terms, impulse is the effect of a force applied over a period of time. For instance, when you hit a stationary cue ball with a cue stick, you are applying a force for a certain duration. This creates an impulse. The unit of impulse is newton-seconds (N·s) and it’s calculated as the product of the force ( F ) and the time ( t ) the force is applied:

- Impulse ( J ) = Force ( F ) × Time ( t )
When a player strikes a cue ball, for example, the impulse transfers to the ball, changing its motion from being stationary to moving with a certain velocity. Thus, the impulse is intimately tied to the change in the ball's momentum.

Momentum can be thought of as the 'quantity of motion' an object possesses. It is a product of the object’s mass and velocity. The greater the momentum, the harder it is to stop the object.

- Momentum ( p ) = Mass ( m ) × Velocity ( v )
In the context of our pool exercise, the pool player imparts an impulse that changes the momentum of the cue ball, initially at rest. According to the impulse-momentum theorem, the change in the momentum of the ball ( Δp ) is equivalent to the impulse imparted:

- Impulse ( J ) = Change in Momentum ( Δp ).
The momentum change can be written as: - Δp = mv_f - mv_i where m is mass, v_f is final velocity, and v_i is initial velocity. Since the cue ball starts from rest, v_i = 0 , simplifying our calculations to Δp = mv_f .

Calculating the final velocity of the cue ball after it receives an impulse involves using the relationship between impulse and momentum change. From the previous discussions, the final velocity (v_f) is obtained by rearranging the impulse-momentum equation for v_f:

- J = 0.25 \, \text{kg} \times v_fGiven:- Impulse (J) = 3.2 N·s- Mass (m) = 0.25 kg
You plug these values into the equation and solve for the final velocity:

\[v_f = \frac{3.2 \, \text{N}\cdot\text{s}}{0.25 \, \text{kg}} = 12.8 \, \text{m/s}\]
So, the speed of the cue ball just after the impact is 12.8 meters per second. This demonstrates how a seemingly simple action, such as striking a ball, involves calculated physics principles to determine outcomes like speed.