### Video Transcript

The spring of a spring gun has a
force constant π 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, weβve called this
distance π. And weβre 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 isnβt 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 π , 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 π times π
times β.

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

Rearranging this expression to
solve for π, we find itβs equal to the square root of two times π times π times β
all divided by π. π, the acceleration due to
gravity, weβll treat as exactly 9.8 meters per second squared. When we plug in for these values,
weβre careful to convert our mass into a value in units of kilograms and our spring
constant π into a value in units of newtons per meter.

Calculating this result, we find π
is 3.5 centimeters. Thatβs the amount the spring must
be compressed in order to fire the ball up to this given height β.