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Kinetic energy is a type of energy that an object possesses due to its motion. It's determined by the equation:

• \( KE = \frac{1}{2} mv^2 \)

Here, \( m \) represents the mass of the object, and \( v \) stands for its velocity. This formula fits perfectly with the scenario of the baseball, as its initial kinetic energy can be calculated knowing its mass (0.140 kg) and its initial speed (40.0 m/s).
To find the exact value, you substitute these numbers into the formula:

- The kinetic energy comes out to be 112.0 J before the bat strikes it.
Remember that kinetic energy depends heavily on the speed – doubling the speed will quadruple the kinetic energy! This is because velocity is squared in the equation. Thus, as the speed of the baseball changes, so does its kinetic energy.

Gravitational potential energy (often referred to simply as potential energy in this context) is the energy an object has due to its position in a gravitational field. This type of energy is calculated using the formula:

• \( PE = mgh \)

In this formula, \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (9.8 m/s² on Earth), and \( h \) is the height above a reference point. For our baseball problem, it's key to understanding how high the ball is positioned after leaving the bat.
Once the ball is 25.0 m above the point of impact, we can calculate its gravitational potential energy to be 34.3 J.

This energy represents the capacity of the baseball to do work as it moves back down to a lower height, and it's subtracted from the total mechanical energy when calculating the final kinetic energy.

Mechanical energy is the sum of kinetic and potential energy in a system. When dealing with problems involving energy, like a baseball flying through the air, mechanical energy is essential because it stays constant (ignoring air resistance) as the ball moves.

In our exercise:

• We start with initial kinetic energy (112.0 J) before the bat hits the ball.
• The bat does additional work on the ball (70.0 J), changing its kinetic energy.
• Finally, as the ball rises to 25.0 m, it gains gravitational potential energy (34.3 J).

Even with a height change, mechanical energy shows how different types of energy convert back and forth. When the work-energy theorem is applied, the mechanical energy principle ensures that the total energy (kinetic plus potential) aligns after considering the exerted work. Therefore, the final energy state of the baseball at 25.0 m lets us calculate the speed post-impact, resulting in a final speed of approximately 45.91 m/s.