# University of Wisconsin Green Bay

A child’s toy that is made to shoot ping pong balls consists of a tube, a spring (k = 18 N/m) and a catch for the spring that can be released to shoot the balls. When a ball is loaded into the tube, it compresses the spring 9.5 cm. If you shoot a ping pong ball straight up out of this toy, how high will it go?

• In this problem, you are asked to find a vertical position of a ball when you are given its initial position on a spring. In both locations, the speed of the ball is zero.

If non-conservative forces are either known or small and if energy is converted from one form to another between the locations, then any time you relate speed and position of an object at two different points, conservation of energy is the most direct way to understand the problem.

In this case, you start out with stored energy in the compression of the spring and convert it to stored gravitational energy.

• Any time you understand the motion of an object by looking at its energy, you begin with the Conservation of Energy equation.

• The ping pong ball rises to a height of 3.1 m above its position on the compressed spring. No further mathematical solution is needed for this problem.

• In this problem, you are asked to find how high a ping pong ball rises when it is shot off of a spring. The energy conversion chain for this motion is

spring potential energy→kinetic energy→kinetic energy + gravitational potential energy→gravitational potential energy

Because essentially no energy is lost from the system (air resistance is negligible,) the amount of energy in the system remains constant throughout. This means energy can be compared at any two points. In this case, we are given information about the energy at the starting point and are asked for information about the top of the motion. In other words,

spring potential energystart = gravitational potential energytop
or
½ kΔx12 = mgh2

as seen in the equation above (fill in 0 for both velocities.) The ball rises to a height at which all of the energy in the system (initially from the spring) has been converted to gravitational potential energy.

There are many description of motion problems which can be solved by either kinematics or energy. Energy is almost always the most efficient way to approach these problems as long as enough information is given. (Energy does not require dividing vectors into components, or doing Second Law problems to find acceleration.) In this case, the changing position of the ball reflected a change in the energy of the system, we are not asked to find time, and there are no significant unknown energy losses from the system. Therefore, we are able to approach this question by tracking the energy.