91Ó°ÊÓ

Impulse is an important concept in physics that can help us understand how forces interact with objects over short periods. When you think about impulse, think about it as the influence that changes an object's momentum.
It happens when a force acts on an object over a certain time. The formula to calculate impulse is given by\[ J = \Delta p = m(v_f - v_i) \]
where:

• \( J \) is the impulse.
• \( m \) is the mass of the object.
• \( v_f \) is the final velocity.
• \( v_i \) is the initial velocity.

In the exercise, the hammer initially has a velocity of 7.5 m/s and then comes to rest, meaning its final velocity is 0 m/s.
This change in velocity results in a change in momentum, providing us with the impulse given to the nail.
Keep in mind the sign of the impulse can indicate direction; in this case, a negative impulse shows that the hammer's velocity is reduced till it stops.

Average force is the constant force that would produce the same change in an object's motion during the same time interval as the actual variable force does.
Knowing the impulse, we can relate it to average force through the formula:\[ J = F_{\text{avg}} \times \Delta t \]
where:

• \( J \) is the impulse.
• \( F_{\text{avg}} \) is the average force.
• \( \Delta t \) is the time interval over which the force acts.

The given problem provides a time interval of 8.0 ms, or 8.0 \( \times \) 10^{-3} seconds.
Using the impulse value of -90 Ns, we can solve for the average force:\[ F_{\text{avg}} = \frac{-90}{8.0 \times 10^{-3}} = -11250 \text{ N} \]
The negative sign tells us that the force's direction is opposite to the direction the hammer was initially moving.

Understanding momentum change is crucial for analyzing collisions and other interactions in physics. Momentum itself is the product of an object's mass and its velocity:\[ p = m \times v \]
The larger the momentum, the harder it is to stop or change the object's direction.
In this exercise, the hammer experiences a momentum change due to a force acting over time.
Here's how it breaks down:

• Initial momentum: The hammer starts with a momentum \( 12 \times 7.5 = 90 \text{ kg m/s} \).
• Final momentum: As the hammer comes to rest, its momentum becomes 0 \text{ kg m/s}.
• Change in momentum: This equals \(-90 \text{ kg m/s}\), the same as the impulse.

Momentum change gives us a clear indication of how much motion has been altered due to a given force, which is why it's directly connected to impulse.