Question

(a) A baseball weighs 5.13 oz. What is the kinetic energy in joules of this baseball when it is thrown by a major-league pitcher at \(95.0 \mathrm{mph}\) ? (b) By what factor will the kinetic energy change if the speed of the baseball is decreased to \(55.0 \mathrm{mph} ?\) (c) What happens to the kinetic energy when the baseball is caught by the catcher? (d) What careful experimental measurement could (in principle) be made to confirm your answer to (c)?

Short answer

The kinetic energy of a 5.13 oz baseball thrown at 95 mph is approximately 131.94 J. When the speed is decreased to 55 mph, the kinetic energy changes by a factor of about 0.328. The kinetic energy is ideally reduced to zero when caught by the catcher. A measurement to confirm this would involve tracking motion and force data to calculate the work done during the catching process, which should be equal to the initial kinetic energy.

Step-by-step solution

(a) Convert units

First, convert the weight of the baseball from ounces to kilograms: 1 oz = 0.0283495 kg, so 5.13 oz = 5.13 × 0.0283495 kg ≈ 0.145 kg. Now, convert the speed from mph to m/s: 1 mph = 0.44704 m/s, so 95 mph = 95 × 0.44704 m/s ≈ 42.47 m/s.

(a) Calculate kinetic energy

Now, we can calculate the kinetic energy using the formula: KE = 0.5 × m × v^2 where KE is the kinetic energy, m is the mass of the baseball (0.145 kg), and v is its speed (42.47 m/s). KE = 0.5 × 0.145 kg × (42.47 m/s)^2 ≈ 131.94 J Thus, the kinetic energy of the baseball is approximately 131.94 joules.

(b) Calculate new kinetic energy

First, convert the new speed from mph to m/s: 55 mph = 55 × 0.44704 m/s ≈ 24.58 m/s. Now, calculate the kinetic energy at the decreased speed: KE_new = 0.5 × 0.145 kg × (24.58 m/s)^2 ≈ 43.34 J

(b) Calculate change factor

Divide the new kinetic energy value by the initial kinetic energy value to determine the factor by which the kinetic energy changes: factor = 43.34 J / 131.94 J ≈ 0.328 The kinetic energy changes by a factor of approximately 0.328.

(c) Kinetic energy when caught

When the baseball is caught by the catcher, its speed decreases rapidly to zero. A decrease in speed results in a decrease in kinetic energy. Therefore, when the baseball is caught, the kinetic energy is ideally reduced to zero.

(d) Experiment to confirm answer

In principle, a careful experimental measurement to confirm the answer to (c) could involve tracking the motion and force of the baseball during and after it is caught by the catcher. This data can be used to calculate the work done by the catcher's hand and glove during the process, which should be equal to the initial kinetic energy of the baseball. If the work done is equal to the initial kinetic energy, it would confirm that the kinetic energy was indeed reduced to zero when the baseball was caught.

Key concepts

Unit Conversion

When dealing with physics problems, different units of measurement are often used.
For the calculations to be accurate, these units need to be converted to standard SI units, like kilograms and meters per second. In this exercise, we first needed to convert the weight of the baseball from ounces to kilograms.
The conversion factor used is: 1 ounce = 0.0283495 kilograms. Therefore, a baseball weighing 5.13 ounces is equivalent to approximately 0.145 kilograms. Similarly, to convert from miles per hour (mph) to meters per second (m/s), the conversion factor is: 1 mph = 0.44704 m/s.
Thus, the speed of the baseball initially thrown at 95 mph converts to approximately 42.47 m/s. This ensures that all calculations are carried out in consistent units, crucial for achieving correct results in physics.

Kinetic Energy Calculation

The kinetic energy of an object is the energy it possesses due to its motion.
It can be calculated using the formula: \[ KE = \frac{1}{2} m v^2 \] Here, \(m\) is the mass in kilograms and \(v\) is the speed in meters per second.
For the baseball exercise, the mass \(m\) is 0.145 kg and the speed \(v\) is 42.47 m/s.Substituting these values in, we find the kinetic energy: \[ KE = 0.5 \times 0.145 \times (42.47)^2 \approx 131.94 \, \text{Joules} \]This computation shows how kinetic energy increases with the square of the velocity, emphasizing why speed greatly affects energy.

Work-Energy Principle

The work-energy principle is a fundamental concept stating that the work done on an object is equal to the change in kinetic energy.
When calculating the factor by which kinetic energy changes if the baseball's speed decreases to 55 mph, we again apply the kinetic energy formula.First, converting 55 mph to m/s gives us 24.58 m/s.
Then, the kinetic energy at this new speed is: \[ KE_{\text{new}} = 0.5 \times 0.145 \times (24.58)^2 \approx 43.34 \, \text{Joules} \]The factor change is determined by dividing the new kinetic energy by the initial kinetic energy: \[ \text{factor} = \frac{43.34}{131.94} \approx 0.328 \]This result indicates that the kinetic energy decreases significantly as the speed of the baseball decreases.

Physics Experiment

In the exercise, the baseball's kinetic energy when caught is reduced to zero because the velocity becomes zero.
To experimentally confirm this, one might measure the forces acting on the baseball as it's caught. The catcher applies a force over a distance to stop the ball, which is equivalent to the work done on the baseball using the work-energy principle. Measurement devices, such as force sensors, could be employed to record the data.
By calculating the work done based on these measurements, it could be determined if it equals the initial kinetic energy of 131.94 joules. This type of experiment not only verifies theoretical calculations but also demonstrates the conservation of energy through practical application.