Problem 15 - University Physics with Modern Physics

Question

To warm up for a match, a tennis player hits the \(57.0-\) g ball vertically with her racket. If the ball is stationary just before it is hit and goes 5.50 \(\mathrm{m}\) high, what impulse did she impart to it?

Step-by-Step Solution

Answer

The impulse imparted to the ball is approximately 0.592 Ns.

Step 1: Understanding the Problem

We are asked to find the impulse imparted to a tennis ball by a player, knowing the initial and final states of the ball.

Step 2: Identifying Known Values

The mass of the tennis ball is given as 57.0 g, which we convert to 0.057 kg. The ball reaches a height of 5.50 m before stopping. It starts from rest.

Step 3: Relating Impulse and Change in Momentum

The impulse imparted to an object is equal to the change in its momentum. Impulse \( J = \Delta p = m \Delta v \). The ball's initial kinetic energy is zero because it starts from rest.

Step 4: Using Energy Conservation to Find Final Velocity

When the ball reaches its maximum height, all of its kinetic energy has been converted to potential energy. We use the equation for gravitational potential energy, \( mgh = \frac{1}{2}mv^2 \), where \( g \) is the acceleration due to gravity (\( 9.81 \, m/s^2 \)).

Step 5: Solving for Final Velocity

Rearranging the energy equation, we get \( v = \sqrt{2gh} \). Substituting \( g = 9.81 \, m/s^2 \) and \( h = 5.50 \, m \), we find \( v = \sqrt{2 \times 9.81 \times 5.50} \approx 10.39 \, m/s \).

Step 6: Calculating the Change in Momentum

The change in momentum is calculated using \( \Delta p = m \Delta v \), where \( \Delta v = v - 0 = 10.39 \, m/s \). Thus, \( \Delta p = 0.057 \, \text{kg} \times 10.39 \, m/s \approx 0.592 \, \text{Ns} \).

Step 7: Determining the Impulse

The impulse imparted to the ball is equal to the change in momentum, \( J = 0.592 \, \text{Ns} \).

Key Concepts

Change in Momentum

When we talk about change in momentum in physics, we're referring to how the velocity of an object, like a tennis ball, changes when a force is applied. Momentum itself is the product of an object's mass and velocity. For the tennis ball, its initial velocity is zero because it starts from a stationary position. The change in momentum is calculated using the formula:

• \( \Delta p = m \Delta v \)
• where \( m \) is the mass and \( \Delta v \) is the change in velocity.

The impulse given to the ball by the tennis player's racket is the same as this change in momentum. This is crucial because impulse lets us understand how quickly the ball moves after being hit. It tells how much oomph, or push, the ball got from the hit.

Energy Conservation

Energy conservation is a fundamental concept in physics, and it's especially helpful when analyzing motion. It tells us that energy cannot be created or destroyed, only transformed from one type to another.When the tennis player hits the ball, the kinetic energy increases as it's converted from the player's effort. As the ball rises, its kinetic energy is gradually converted into gravitational potential energy. Hence, at any point:

• The total energy of the ball remains constant.
• The initial kinetic energy is transformed completely into potential energy at the ball's highest point.

In this case, the kinetic energy (\( KE = \frac{1}{2} mv^2 \)) is converted into gravitational potential energy (\( PE = mgh \)). Understanding energy conservation helps us track these changes and predict the ball's behavior through its flight.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy stored in an object because of its height above the ground. The higher an object is, the more gravitational potential energy it has.For the tennis ball, as it reaches its highest point of 5.50 meters, all the kinetic energy imparted by the racket is transformed into gravitational potential energy. The formula for this energy is:

• \( PE = mgh \)
• where \( m \) is the mass, \( g \) is the acceleration due to gravity (approximately \( 9.81 \, m/s^2 \)), and \( h \) is the height.

This transformation is what eventually stops the ball as it reaches the peak of its flight. Gravitational potential energy is a key concept for understanding how objects move under the influence of gravity.