Example

Question:

To simulate car accidents, a car of mass \(1000\,\mathrm{kg}\) is moving with a speed \(18.0\,\mathrm{km/h}\) (\(5\,\mathrm{m/s}\)) on a smooth road and collides with a horizontally mounted spring of spring constant \(5.25 \times 10^3\,\mathrm{N\,m^{-1}}\). What is the maximum compression of the spring?

Solution:

At maximum compression, the car's kinetic energy is completely converted to the spring's potential energy.
Kinetic energy of car: \[ K = \frac{1}{2}mv^2 = \frac{1}{2} \times 1000 \times 5^2 = 1.25 \times 10^4\,\mathrm{J} \] At maximum compression \(x_m\), \[ V = \frac{1}{2}k x_m^2 = K \] \[ \frac{1}{2} \times 5.25 \times 10^3 \times x_m^2 = 1.25 \times 10^4 \] \[ x_m^2 = \frac{2 \times 1.25 \times 10^4}{5.25 \times 10^3} = \frac{2.5 \times 10^4}{5.25 \times 10^3} \approx 4.76 \] \[ x_m = 2.00\,\mathrm{m} \] Assumptions: Spring is massless, surface has negligible friction.