91Ó°ÊÓ

The impulse-momentum theorem is a fundamental principle that connects impulse and a change in momentum. Here, impulse refers to the effect of a force acting over a period of time. We denote it as \[ J \].

• Impulse, \( J \), can be expressed as the product of the average force, \( F_{avg} \), and the time duration, \( t \), the force acts: \( J = F_{avg} \cdot t \).
• Momentum, on the other hand, relates to the mass and velocity of an object, and is given by \( p = m \times v \).

According to the impulse-momentum theorem, the impulse on an object is equal to the change in its momentum. This can be expressed mathematically as:\[ J = \Delta p = p_f - p_i \]Here, \( p_f \) is the final momentum and \( p_i \) is the initial momentum. In our exercise, the skater's initial momentum is zero since she starts from rest. Her final momentum is calculated by \( p_f = m \cdot v = 46 \times 1.2 \), resulting in \( 55.2 \) kg·m/s. By the theorem, the impulse exerted equals this change in momentum.

Newton's Third Law of Motion is often summarized by the statement: "For every action, there is an equal and opposite reaction." This means that whenever one object exerts a force on a second object, the second object simultaneously exerts a force of equal magnitude and in the opposite direction back on the first object. In the context of our skater:

• When she pushes off the wall, she exerts a force on the wall.
• In turn, the wall exerts an equal and opposite force back on her.
• Due to this equal and opposite force pair, the skater is propelled backward, while the force the wall feels is in the opposite direction.

It's important to remember that these forces are equal in size but opposite in direction. This core concept explains why, despite the skater's exertion of force on the wall, she is able to propel herself away with a velocity of 1.2 m/s.

Average force provides a simplified representation of the force that causes an object to change its momentum over a given time period. It helps us to understand the effect of forces that are not constant over time. For the skater, who applies a force for 0.80 seconds:

• The average force is calculated by distributing the impulse over the time of contact: \( F_{avg} = \frac{\Delta p}{t} \).
• With the change in momentum \( \Delta p = 55.2 \) kg·m/s and the time of contact \( t = 0.80 \) s, we find \( F_{avg} = \frac{55.2}{0.80} \).
• This calculation results in an average force \( F_{avg} = 69 \) N.

The direction of this average force is noteworthy: according to Newton's third law, it acts in the same direction as the wall's reaction force—opposite to the skater's motion. This understanding of average force not only aids in calculations but also strengthens our grasp of dynamic interactions in the real world.