Question

Think \& Calculate A pitcher accelerates a 0.14-kg hardball from rest to \(25.5 \mathrm{~m} / \mathrm{s}\) in \(0.075 \mathrm{~s}\). (a) How much work does the pitcher do on the ball? (b) What is the pitcher's power output during the pitch? (c) Suppose the ball reaches \(25.5 \mathrm{~m} / \mathrm{s}\) in less than \(0.075 \mathrm{~s}\). Is the power produced by the pitcher in this case more than, less than, or the same as the power found in part (b)? Explain.

Short answer

(a) 45.52 J; (b) 606.9 W; (c) More power, as power is work divided by time.

Step-by-step solution

Calculate the Final Kinetic Energy

The work done on the ball is equal to the change in its kinetic energy. The initial kinetic energy is 0 because the ball starts from rest. The final kinetic energy can be calculated using the formula: \[ KE = \frac{1}{2} m v^2 \]where \( m = 0.14 \, \text{kg} \) is the mass of the ball and \( v = 25.5 \, \text{m/s} \) is the final velocity. Substituting these values, we get:\[ KE = \frac{1}{2} \times 0.14 \, \text{kg} \times (25.5 \, \text{m/s})^2 = 45.5175 \, \text{J} \]

Calculate the Work Done

The work done by the pitcher on the ball is equal to the change in kinetic energy. Since the initial kinetic energy is zero, the work done is equal to the final kinetic energy of the ball:\[ W = 45.5175 \, \text{J} \]

Calculate the Power Output

Power is the rate at which work is done. It can be calculated using the formula:\[ P = \frac{W}{t} \]where \( W = 45.5175 \, \text{J} \) is the work done and \( t = 0.075 \, \text{s} \) is the time taken. Substituting these values, we get:\[ P = \frac{45.5175 \, \text{J}}{0.075 \, \text{s}} = 606.9 \, \text{W} \]

Analyze the Effect of Reduced Time on Power

If the ball reaches the same velocity in less than \( 0.075 \, \text{s} \), the work done remains the same (since the change in velocity and thus kinetic energy is unchanged). However, reducing the time with the same amount of work increases the power, since power is work divided by time. Hence, if the time decreases, the power output increases.

Key concepts

Understanding Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is a fundamental concept in physics that helps us understand how objects move and interact. In simple terms, when an object is in motion, it has kinetic energy. The faster it moves, the more kinetic energy it has.

To calculate kinetic energy (\( KE \)), we use the formula:

• \( KE = \frac{1}{2} m v^2 \)

Here, \( m \) represents the mass of the object, and \( v \) is its velocity. The kinetic energy formula shows that the energy is directly proportional to the mass and the square of the velocity, highlighting the impact of velocity on kinetic energy. If the velocity of an object doubles, its kinetic energy increases by a factor of four!
Understanding kinetic energy is crucial because it connects to how work is done on an object. As in the case of the baseball, when the pitcher throws the ball, he does work to accelerate it, increasing its kinetic energy.

The Basics of Power Calculation

Power in physics refers to the rate at which work is done or energy is transferred over time. It's an essential measure in understanding how efficient an action is. In our baseball example, calculating the pitcher's power gives us insight into how quickly he can transfer energy to the ball.

The formula to calculate power (\( P \)) is as follows:

• \( P = \frac{W}{t} \)

where \( W \) is the work done, and \( t \) is the time during which the work is done. This equation shows the relationship between work and the time taken to do that work. If the same amount of work is done in less time, the power increases. Power is typically measured in watts, where one watt is equivalent to one joule per second.

In practical terms, when the pitcher throws the ball in the given time period, we can see how powerful the pitch is by looking at the watts. If the ball reaches its speed faster, the pitcher's power increases because he's doing the same amount of work in a shorter duration.

Exploring the Work-Energy Principle

The work-energy principle is a fundamental concept in physics that establishes the connection between work and changes in kinetic energy. It states that the work done by all the forces acting on an object is equal to the change in its kinetic energy. Put simply, when work is done on an object, it results in a change in the object's kinetic energy.This principle is beautifully illustrated in the pitcher's scenario. As the pitcher throws the ball, he does a certain amount of work, and this work translates into the ball's kinetic energy. Initially, the ball is at rest, and its kinetic energy is zero. Once the pitcher exerts force over a distance and accelerates the ball, the work done transforms into kinetic energy.

Using the work-energy principle, we can derive the work done as:

• \( W = \Delta KE = KE_{final} - KE_{initial} \)

Since the ball starts from rest, its initial kinetic energy is zero, and thus the work done is equal to the final kinetic energy of the ball.
This principle not only helps in analyzing motion scenarios but also in designing and understanding mechanical systems where energy transformation is a key aspect.