Problem 9 - College Physics

Question

A \(0.280-\mathrm{kg}\) volleyball approaches a player horizontally with a speed of \(15.0 \mathrm{~m} / \mathrm{s}\). The player strikes the ball with her fist and causes the ball to move in the opposite direction with a speed of \(22.0 \mathrm{~m} / \mathrm{s}\). (a) What impulse is delivered to the ball by the player? (b) If the player's fist is in contact with the ball for \(0.0600 \mathrm{~s}\), find the magnitude of the average force exerted on the player's fist.

Step-by-Step Solution

Answer

The impulse delivered to the ball by the player is -10.36 N.s. The average force exerted on the player's fist is -172.6 N.

Step 1: Calculate the Impulse

To determine the impulse delivered to the ball by the player, apply the formula for impulse, which is the change in momentum of the ball. Given that mass, m = 0.280 kg, the initial velocity, \( v_{i} = 15.0 m/s \), and the final velocity, \( v_{f} = -22.0 m/s \) (the negative sign indicates a change in direction), calculate the impulse, \( Impulse = m (v_{f} - v_{i}) = 0.280 kg (-22.0 m/s - 15.0 m/s) \).

Step 2: Calculate the Force

After calculating the impulse, determine the force exerted by the player during the impact period. Given that the time duration, \( \Delta t \), is 0.0600 s, apply the formula \( Force = \frac{Impulse}{\Delta t} \) to get the answer.

Key Concepts

Newton's Laws of Motion

Understanding Newton's Laws of Motion is crucial when analyzing how forces affect the motion of objects, which is exactly what happens when the direction of a volleyball changes due to a player's strike. Newton's First Law states that an object will remain at rest or continue to move at a constant velocity unless acted upon by a force.
This is why the volleyball continues to sail toward the player until a force is applied to change its velocity.
Newton's Second Law further extends this understanding by introducing the relationship between force, mass, and acceleration. It is mathematically expressed as: \[ F = ma \]where \( F \) is the force applied, \( m \) is the mass of the object, and \( a \) is the acceleration.
When the player hits the ball, an acceleration occurs, altering the ball's velocity.
This aspect helps us understand how the force applied by the player's fist contributes to the significant change in the ball’s speed and direction.Finally, Newton's Third Law states that for every action, there is an equal and opposite reaction.
This means that while the player exerts a force on the ball, the ball exerts an equal and opposite force back on the player's fist.
This is elementary in explaining the interaction between the player's forceful hit and the resulting motion of the ball.

Conservation of Momentum

Momentum describes an object's motion and is defined as the product of its mass and velocity, \( p = mv \).
A fascinating aspect of momentum is its conservation in isolated systems unless acted upon by external forces.
In this volleyball problem, we see momentum before and after the ball is struck by the player.
Initially, the ball has momentum due to its mass and incoming velocity.However, once the player strikes the ball, the momentum needs to be recalculated because the velocity changes.
But it's critical to note that the overall momentum in the isolated system of just the volleyball (excluding considerations like air resistance and friction) will remain constant.
This is due to the law of conservation of momentum, which states:

• The total momentum of a closed system is constant.
• The momentum before and after an interaction can be different for individual parts, like the volleyball and player's hand, but the sum total remains the same.

Thus, the interaction results in a change in motion direction but adheres to the overarching principle of conserving total momentum in the system.

Average Force

Calculating the average force helps to understand how much force was applied on average short period of interaction. This is particularly useful when the force isn't constant, as in the instance of the player's fist striking the ball.
Average force can be determined by dividing the impulse by the time duration the force is applied, expressed as: \[ F_{avg} = \frac{Impulse}{\Delta t} \]In our scenario, the impulse was obtained by calculating the change in momentum of the ball upon being struck.
This impulse was then divided by the time the player's fist was in contact with the ball to get the average force.
Understanding average force in this example helps students decipher how a seemingly short interaction—a fraction of a second—can have such a profound effect on the movement of the volleyball. It shows how quickly and effectively the force was applied to alter the ball's trajectory so dramatically.