There are two main formulas for modeling the behavior of a spring. The force is found from

F = –kx

Where “k” is the force constant for the spring. It is measured in N/m. “x” is the displacement from equilibrium in meters. “F” is te force in Newtons. The negative sign is in this equation because the force always opposes the direction the spring is stretched. When calculating the magnitude of the force, drop the negative sign. A graph of force versus displacement looks like the one below.

This means the force is not constant but varies with time. This also means that the acceleration changes as the spring’s displacement changes. You cannot use the typical kinematics formulas wtihout taking into account the change acceleration. This is the jerk. To calculate the MAXIMUM acceleration of an object stuck on the spring look at the spring when the spring is compressed the most and set this equal to Newton’ second law.

F_spring = F_Newton

k(x_max) = ma_max

“k” is the spring constant and x_max is the distance the springis compressed, “m” is the train’s mass and a_max is the acceleration of the train. If the final acceleration is zero, once it is released from the spring, then the average acceleration is half the max. Your project limits the avearge acceleration to 5.5 g’s, (52.9 ^m/s²). This means the maximum acceleration is 11 g’s, which is 107.8 ^m/s². Therefore,

k(x_max) = m(107.8 ^m/s²)

You, the designer, need to pick a combination of displacement and spring constant that meets the constraints above. Then put these numbers in the energy equations for other calculations.

When an object hits the spring, the spring removes the energy from body. The energy stored in the spring comes from finding the work under the curve on the graph at the top of the page.

This is a potential energy, just like gravitational potential energy. It can be equated to kinetic energy.

The displacement comes from the earlier calculation. If you need more energy, redo your earlier numbers and make the displacement and greater and the spring constant smaller. This will give you more energy because the displacement in the energy formula is squared and the spring constant is not.

Example

A toy car of mass 2.0 kg is traveling towards a spring at 5 m/s. What is the length of the spring and spring constant will slow the toy down with an average acceleration less than or equal to 5 m/s.

Solution The car will experience an range of accelerations. It is zero the instant if touches the spring without compressing it (when x = 0.). The acceleration is at a maxumum when the the spring is compressed as much as possible (when x is at a maximum.)