A car’s bumper is designed to withstand a \(4.0 - {\rm{km}}/{\rm{h}}\) \(\left( {1.1 - {\rm{m}}/{\rm{s}}} \right)\) collision with an immovable object without damage to the body of the car. The bumper cushions the shock by absorbing the force over a distance. Calculate the magnitude of the average force on a bumper that collapses \(0.200{\rm{ m}}\) while bringing a \(900 - {\rm{kg}}\) car to rest from an initial speed of \(1.1{\rm{ m}}/{\rm{s}}\).

Work energy theorem: According to the work-energy theorem, the work done of the body equals the change in kinetic energy of the body.

Mathematically,

\(W = K{E_f} - K{E_i}\)

According to the work-energy theorem,

\(\begin{aligned}W = \frac{1}{2}m{v^2} - \frac{1}{2}m{u^2}\\Fs = \frac{1}{2}m{v^2} - \frac{1}{2}m{u^2}\end{aligned}\)

Here,\(F\)is the force exerted on the bumper,\(m\)is the mass of car\(\left( {m = 900{\rm{ kg}}} \right)\),\(v\)is the final velocity of the car (\(v = 0\)as the car stops after the collision), u is the initial velocity of the car\(\left( {u = 1.1{\rm{ m}}/{\rm{s}}} \right)\), and\(s\)is the distance covered by the car before it comes to rest after collision\(\left( {s = 0.2{\rm{ m}}} \right)\).

The magnitude of the force exerted on the bumper during collision is,

\(F = \left| {\frac{{m\left( {{v^2} - {u^2}} \right)}}{{2s}}} \right|\)

Putting all known values,

\(\begin{aligned}F = \left| {\frac{{\left( {900{\rm{ kg}}} \right) \times \left( {{{\left( 0 \right)}^2} - {{\left( {1.1{\rm{ m}}/{\rm{s}}} \right)}^2}} \right)}}{{2 \times \left( {0.2{\rm{ m}}} \right)}}} \right|\\ = 2722.5\;{\rm{N}}\end{aligned}\)

Therefore, the required force exerted on bumper is \(2722.5\;{\rm{N}}\).