# Conservation of Energy in Projectile Motion: Examples & Analysis

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• 0:04 Free-Fall Conservation…
• 2:11 What is Projectile Motion?
• 3:10 Example Problem
• 5:29 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

A projectile moves through the air with only the force of gravity acting on it, so the total energy of the projectile doesn't change. In this lesson, learn more about how to apply the law of conservation of energy to projectile motion.

## Free-Fall Conservation of Energy

If you throw a ball straight up in the air with an initial velocity of 10 m/s, how high will it go before it turns around and falls back to the ground?

One way to answer this question is by looking at how the energy of the ball changes as it goes up and then comes back down. There are two main kinds of energy that are important in this situation: kinetic and potential. Kinetic energy (K) is the energy of motion, with the equation (as you can see below) being:

This is essentially saying that all moving objects have kinetic energy, and the amount of kinetic energy is proportional to both how fast the object is moving (velocity, v) and how heavy it is (mass, m).

In contrast to kinetic energy, an object can have gravitational potential energy whether it's moving or not. The amount of gravitational potential energy (UG) of the ball depends on how high it is above the ground (h) and its weight (W = mg, where g = 9.8 N/kg), with the equation, as you can see below, being:

If no forces (other than gravity) act on the ball during its trip up and back down, then the ball's total mechanical energy (kinetic + potential) will not change. However, the energy of the ball can change forms. Initially, the ball has a lot of kinetic energy, because you've given it a pretty large upward velocity. In this case, the total energy is conserved because it doesn't change.

However, as the ball moves upward, it slows down as its initial kinetic energy is transformed into potential energy. Eventually, all of the initial kinetic energy will become potential energy, and the ball will stop momentarily. The ball has finally reached its highest point. After this, it'll turn around and fall back the ground as all the potential energy it had at the highest point is transformed back into kinetic energy as it falls to the ground.

This change in energy can be represented using a bar chart that shows how much kinetic and potential energy the ball has at different times. Notice that the total energy is the same in both cases, but just after the ball is thrown, all its energy is kinetic. When it reaches the maximum height, all the energy has now been converted into potential energy.

## What Is Projectile Motion?

How would our analysis of the ball's motion change if it was not thrown straight up? What if the ball was thrown at an angle instead? When an object is moving through the air in both the horizontal and vertical directions, we call this projectile motion. Even though the motion is a little different than the motion of a ball that's thrown straight up, you can still determine how high the ball will go by examining how its energy changes.

Because gravity is still the only force acting on a projectile, the total energy will still not change. However, what's different this time is that the ball never reaches a point where it stops moving, even for an instant. When it gets to its maximum height, it's still moving forward, even though its vertical velocity is 0, so it still has some kinetic energy. Let's see what that would look like on a graph like the one below:

As you can see, when a ball is thrown at an angle, it never reaches a point where its kinetic energy is 0. However, some of its kinetic energy does become potential energy, and potential energy is at a maximum when the ball reaches its highest point.

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