Question

Calculate the kinetic energy of a baseball (mass \(=5.25 \mathrm{oz}\) ) with a velocity of \(1.0 \times 10^{2} \mathrm{mi} / \mathrm{h}\).

Short answer

The kinetic energy of the baseball is \(148.5 Joules\).

Step-by-step solution

Convert mass from ounces to kilograms

To convert the mass of the baseball from ounces to kilograms, we can use the conversion factor: 1 oz = 0.0283495 kg We have the mass given as 5.25 oz. So, mass = 5.25 oz * 0.0283495 kg/oz = 0.148835 kg.

Convert velocity from miles/hour to meters/second

To convert the velocity of the baseball from miles per hour to meters per second, we can use the conversion factors: 1 mile = 1609.34 meters 1 hour = 3600 seconds We have the velocity given as 100 mi/h. So, velocity = 100 mi/h * (1609.34 m/mi) * (1 h/3600 s) = 44.704 m/s

Apply the kinetic energy formula

Now that we have the proper SI units for mass and velocity, we can use the Kinetic Energy formula: KE = (1/2) * mass * velocity^2

Calculate the kinetic energy of the baseball

Substituting the values for mass and velocity, we have: KE = (1/2) * 0.148835 kg * (44.704 m/s)^2 Calculating the result: KE = 0.0744175 kg * 1999.51 m^2/s^2 = 148.5 J The kinetic energy of the baseball is 148.5 Joules.

Key concepts

Kinetic Energy Formula

Kinetic energy is a term that describes the energy an object possesses due to its motion. It's a concept that shows up often in physics problems and understanding how to calculate it is essential for students. The formula for kinetic energy (KE) is expressed as:
\[\begin{equation} KE = \frac{1}{2} \times mass \times velocity^2 \end{equation}\]
Mass here refers to the object's mass and velocity is the speed of the object in a specific direction. The factor of \(\frac{1}{2}\) shows up because kinetic energy is derived from the work done to accelerate an object from rest to its current velocity, which involves an average factor of the starting and ending velocity when starting from rest, which is \(\frac{0 + v}{2} = \frac{1}{2}v\). To calculate an object's KE, you just need to plug its mass and velocity into this formula. This calculation yields the kinetic energy in Joules (J) in the metric system, which is the standard unit of energy.

Unit Conversion

When dealing with physics problems, unit conversion often becomes essential because the International System of Units (SI) is the standard used in scientific calculations. However, measurements may not always be provided in SI units. Just like in the given exercise, you might find mass in ounces (oz) and velocity in miles per hour (mph). Before applying the kinetic energy formula, you must convert these values into kilograms (kg) for mass and meters per second (m/s) for velocity.
Unit conversion involves multiplying the given measurement by a conversion factor that equals 1. For instance, since 1 oz is equal to 0.0283495 kg, you can convert ounces to kilograms as follows:
\[\begin{equation} mass_{\text{(kg)}} = mass_{\text{(oz)}} \times 0.0283495 \frac{\text{kg}}{\text{oz}} \end{equation}\]
Similarly, to convert mph to m/s, you would use:
\[\begin{equation} velocity_{\text{(m/s)}} = velocity_{\text{(mph)}} \times 1609.34 \frac{\text{m}}{\text{mi}} \times \frac{1}{3600} \frac{\text{h}}{\text{s}} \end{equation}\]
Understanding these conversions is critical, not only for achieving the correct answer but also for developing a deeper comprehension of the relationship between different units of measure.

Velocity and Mass Relationship

The kinetic energy of an object is directly related to both its mass and the square of its velocity, as indicated by the kinetic energy formula. This means that even a small increase in velocity will have a significant effect on the kinetic energy because of the velocity squared term.
If the velocity of an object doubles, for example, its kinetic energy will increase by a factor of four (since \(2^2 = 4\)). This relationship shows why high speeds are such a critical factor in safety considerations, such as vehicle crash energy transfer. On the other hand, mass also plays a crucial role. A more massive object with the same velocity will have greater kinetic energy due to its larger mass. Hence, an understanding of this relationship is not only important for solving kinetic energy problems but also for realizing how different factors affect an object's energy in motion.