# Video: Determining the Spring Compression Required to Launch a Projectile to a Particular Height

The spring of a spring gun has a force constant 𝑘 = 12 N/cm. When the gun is aimed vertically upward, a 15-g projectile is shot to a height of 5.0 m above the end of the expanded spring, as shown in the accompanying diagram. How much was the spring compressed initially?

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### Video Transcript

The spring of a spring gun has a force constant 𝑘 equals 12 newtons per centimeter. When the gun is aimed vertically upward, a 15-gram projectile is shot to a height of 5.0 meters above the end of the expanded spring, as shown in the accompanying diagram. How much was the spring compressed initially?

In the diagram, we’ve called this distance 𝑑. And we’re also told that the spring constant of the spring is 12 newtons per centimeter and the mass of the shot being fired is 15 grams. Looking at the diagram of this scenario, we see that the initial and final conditions are connected through energy conservation.

If we call the state where the spring is compressed the initial state and the point where the ball is at its highest point in flight the final state, then we can write that the initial sum of kinetic plus potential energy is equal to the final amount of kinetic plus potential energy. Both at the start and at the finish, the ball isn’t in motion, so its kinetic energy in both those cases is zero.

When we consider the initial state of the ball, we know that its potential energy is due to the elastic energy compressed in the spring. We can call this PE sub 𝑠, for spring potential energy. On the other hand, the final potential energy of our system is entirely gravitational as the ball is motionless a known height above the expanded spring. So, we can write that spring potential energy equals gravitational potential energy. And we recall that spring potential energy is equal to one-half the spring constant times the displacement from equilibrium squared and that gravitational potential energy is equal to 𝑚 times 𝑔 times ℎ.

For our scenario, we write the spring potential energy is one-half 𝑘 times 𝑑 squared, where 𝑑 is the distance we want to solve for. And on the potential energy side, we let the height ℎ be equal to 5.0 meters, the height of the ball when it’s above the fully extended spring and not in motion.

Rearranging this expression to solve for 𝑑, we find it’s equal to the square root of two times 𝑚 times 𝑔 times ℎ all divided by 𝑘. 𝑔, the acceleration due to gravity, we’ll treat as exactly 9.8 meters per second squared. When we plug in for these values, we’re careful to convert our mass into a value in units of kilograms and our spring constant 𝑘 into a value in units of newtons per meter.

Calculating this result, we find 𝑑 is 3.5 centimeters. That’s the amount the spring must be compressed in order to fire the ball up to this given height ℎ.