Question

(a) A baseball weighs \(5.13 \mathrm{oz}\). What is the kinetic energy, in joules, of this baseball when it is thrown by a major-league pitcher at \(95.0 \mathrm{mi} / \mathrm{h}\) ? (b) By what factor will the kinetic energy change if the speed of the baseball is decreased to \(55.0 \mathrm{mi} / \mathrm{h}\) ? (c) What happens to the kinetic energy when the baseball is caught by the catcher?

Short answer

The kinetic energy of the baseball when thrown at 95.0 mi/h is 132.75 J. When the speed is decreased to 55.0 mi/h, the kinetic energy decreases by a factor of approximately 0.327. When the baseball is caught by the catcher, its kinetic energy becomes zero.

Step-by-step solution

Convert units

First, let's convert the mass of the baseball from ounces (oz) to kilograms (kg). 1 oz ≈ 0.0283 kg, so: \(5.13 \ oz = 5.13 * 0.0283 = 0.145 \ kg\) Now we need to convert the speed of the baseball from miles per hour (mi/h) to meters per second (m/s). 1 mi/h ≈ 0.447 m/s, so: \(95.0 \ mi/h = 95.0 * 0.447 = 42.47 \ m/s\) \(55.0 \ mi/h = 55.0 * 0.447 = 24.59 \ m/s\) #a#:

Calculate the kinetic energy

Now that we have the mass in kilograms, and the speed in meters per second, we can use the kinetic energy formula to find the kinetic energy: \(KE_{95} = \frac{1}{2} \times 0.145 \times (42.47)^2 = 132.75 \ J\) The kinetic energy of the baseball when thrown at 95.0 mi/h is 132.75 J. #b#:

Calculate the change in kinetic energy

Now, let's find the kinetic energy of the baseball at a speed of 55.0 mi/h: \(KE_{55} = \frac{1}{2} \times 0.145 \times (24.59)^2 = 43.44 \ J\) Now, we will find the factor by which the kinetic energy changes by dividing the kinetic energies: \(\text{Factor} = \frac{KE_{55}}{KE_{95}} = \frac{43.44}{132.75} = 0.327\) (approximately) When the speed of the baseball is decreased to 55.0 mi/h, the kinetic energy decreases by a factor of approximately 0.327. #c#:

Kinetic energy when the baseball is caught

When the baseball is caught by the catcher, its speed reduces to zero. As the velocity becomes zero, the kinetic energy of the baseball also becomes zero.

Key concepts

Unit Conversion

When dealing with physical quantities, one of the essential skills to master is unit conversion. This is because units vary across different measurement systems, like from ounces to kilograms or miles per hour to meters per second. For kinetic energy calculations, the standard units are kilograms for mass and meters per second for velocity. To convert from ounces (oz) to kilograms (kg), you can use the conversion factor: 1 oz ≈ 0.0283 kg. Similarly, to convert from miles per hour (mi/h) to meters per second (m/s), the conversion factor is: 1 mi/h ≈ 0.447 m/s.

Converting to the standard SI units is crucial because it allows for consistency in calculations and makes it possible to use the kinetic energy formula accurately. Remember that if you skip or make an error in this step, it could lead to incorrect results, irrespective of how well you perform the subsequent steps of the calculation.

Kinetic Energy Formula

The kinetic energy (\( KE \)) of an object in motion is given by the formula \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. This relationship indicates that kinetic energy is directly proportional to the mass of the object and the square of its velocity. Because of the squaring of the velocity, slight changes in speed can lead to significant changes in kinetic energy.

Using this formula requires that you have the mass in kilograms and the speed in meters per second. Once you have the correct units, as achieved through unit conversion, the calculation becomes straightforward. By substituting the values of mass and velocity into the formula, you can solve for the kinetic energy.For instance, taking the provided example of a baseball, after unit conversion, you input the mass and velocity and perform the equation to find that a baseball thrown by a major-league pitcher at 95.0 mi/h has a kinetic energy of 132.75 joules.

Mass and Velocity Relationship

The relationship between mass and velocity in kinetic energy is straightforward yet profound. The mass (\( m \) correlates linearly with kinetic energy, meaning if you double the mass while keeping velocity constant, the kinetic energy also doubles. However, velocity (\( v \)) has a squared relationship with kinetic energy. Consequently, a small change in velocity can lead to a proportionally larger change in kinetic energy.

For example, when a baseball's speed is reduced from 95.0 mi/h to 55.0 mi/h, the kinetic energy drops significantly, by a factor of approximately 0.327. This shows how sensitive kinetic energy is to the velocity of an object. Additionally, in the case where the baseball is caught and its velocity drops to zero, the kinetic energy also drops to zero, regardless of the mass, because any number multiplied by zero is zero.