Question

A \(0.15-\mathrm{kg}\) baseball has a kinetic energy of \(18 \mathrm{~J}\). What is its speed?

Short answer

The speed of the baseball is approximately 15.49 m/s.

Step-by-step solution

Understand Kinetic Energy Formula

Kinetic energy \( KE \) is given by the formula: \[KE = \frac{1}{2}mv^2\]where \( m \) is mass and \( v \) is velocity. We know the kinetic energy \( KE = 18 \, \mathrm{J} \) and the mass \( m = 0.15 \, \mathrm{kg} \). We need to solve for \( v \).

Rearrange the Formula

Rearrange the kinetic energy formula to solve for \( v \):\[v^2 = \frac{2 \, KE}{m}\]

Substitute Known Values

Substitute the known values into the rearranged formula:\[v^2 = \frac{2 \times 18}{0.15}\]

Calculate

Calculate \( v^2 \):\[v^2 = \frac{36}{0.15} = 240\]Then, calculate \( v \) by taking the square root of \( 240 \):\[v = \sqrt{240} \approx 15.49 \, \mathrm{m/s}\]

Check Units & Solution

Ensure that the units for kinetic energy, mass, and speed are consistent and correct. The speed \( v \) is in meters per second \( \mathrm{m/s} \), which matches the required unit of speed in physics problems.

Key concepts

Physics and Its Principles

Physics is a fascinating field that studies the nature and properties of matter and energy. It is one of the fundamental sciences that helps us understand how the universe operates.
In the context of kinetic energy, physics allows us to explore the relationships between different physical quantities, such as mass, speed, and energy. This involves understanding how energy is stored, transferred, and transformed.

Understanding kinetic energy specifically is essential, as it is the energy possessed by an object due to its motion. The greater the mass and speed of an object, the more kinetic energy it has.
The kinetic energy of an object can be calculated using the formula:

• \[ KE = \frac{1}{2}mv^2 \]
• Where \( KE \) is the kinetic energy in joules, \( m \) is the mass in kilograms, and \( v \) is the velocity in meters per second.

By rearranging this formula, we can solve for any unknown quantity if the other two are known. This ability to manipulate and understand formulas is a cornerstone of physics problem-solving.

Velocity and Its Significance

Velocity is a crucial concept in physics that defines the speed of something in a specific direction. It differs from speed, which doesn't include direction. Velocity is a vector quantity, meaning it has both magnitude and direction.

In our example of the baseball, velocity determines how fast the baseball moves and its trajectory. When we calculate the kinetic energy of the baseball, we use the speed part of its velocity.
The formula

• \[ v^2 = \frac{2 \, KE}{m} \]

allows us to find the velocity of an object if we know its kinetic energy and mass.
Velocity is not just about how fast an object goes; it also influences other physical phenomena, like momentum. Understanding velocity can help predict the outcomes of collisions or other interactions in motion-based systems.

In practical terms, velocity helps us calculate how an object needs to move to reach a certain point, or how it interacts with forces acting upon it. This makes it an indispensable part of physics and everyday life.

Baseball and Its Energetic Dynamics

Baseballs are small yet dynamic objects often used in physics examples due to their properties. With a standard mass and a known size, baseballs can effectively demonstrate principles of motion, energy, and force.
In baseball, kinetic energy is especially vital. As players throw or hit the ball, they transfer energy into the ball, giving it speed and momentum.
To calculate the speed of a baseball when given its kinetic energy and mass, you can use the rearranged kinetic energy formula discussed earlier. Knowing the mass is \( 0.15 \, \mathrm{kg} \) and kinetic energy is \( 18 \, \mathrm{J} \), we can find its velocity to be around \( 15.49 \, \mathrm{m/s} \).

This speed is essential for players to judge where the ball will end up and how quickly they need to react. It also helps in designing sports equipment, understanding injury mechanics, and strategizing game plans.
Physics and baseball intertwine in various ways, from calculating the ball's trajectory to understanding its impact forces. These calculations help make the game more scientific and enjoyable, and demonstrate the practical application of physics.