Question

(II) At an accident scene on a level road, investigators measure a car's skid mark to be 78 m long. It was a rainy day and the coefficient of friction was estimated to be 0.30. Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes. (Why does the car's mass not matter?)

Short answer

The speed of the car is \(21{\rm{ m/s}}\) when the driver slams on the brakes. The velocity is not the function of the mass of the car, so the car’s mass does not matter.

Step-by-step solution

Given data

The final speed of the car is\(v = 0\).

The kinetic friction coefficient is\({\mu _{\rm{k}}} = 0.30\).

The car stopped after a distance \(d = 78\;{\rm{m}}\).

Calculation of force of friction

Let\({v_{\rm{o}}}\)be the speed of the car when the driver slammed on the brakes.

The normal force on the car is \(N = mg\).

Thus, the friction force on the car is:

\(\begin{array}{c}F = {\mu _{\rm{k}}}N\\ = {\mu _{\rm{k}}}mg\end{array}\)

Application of energy conservation principle to calculate the speed

From the energy conservation principle, the loss in the kinetic energy of the car is equal to the work done by the friction force on the car.

Now, the change in kinetic energy is \(\left( {\frac{1}{2}m{v^2} - \frac{1}{2}mv_{\rm{o}}^2} \right)\).

The work done by the friction force is \(Fd\cos {180^ \circ }\) as the applied force is opposing the motion.

Now, from the work-energy theorem, you can write:

\(\begin{array}{c}\frac{1}{2}m{v^2} - \frac{1}{2}mv_{\rm{o}}^2 = Fd\cos {180^ \circ }\\\frac{1}{2}m{v^2} - \frac{1}{2}mv_{\rm{o}}^2 = {\mu _{\rm{k}}}mgd\cos {180^ \circ }\\\left( {\frac{1}{2}m \times {0^2}} \right) - \frac{1}{2}mv_{\rm{o}}^2 = 0.30 \times m \times \left( {9.80\;{\rm{m/}}{{\rm{s}}^2}} \right) \times \left( {78\;{\rm{m}}} \right)\cos {180^ \circ }\\{v_{\rm{o}}} = 21.42\;{\rm{m/s}}\\{v_{\rm{o}}} \approx 21\;{\rm{m/s}}\end{array}\)

Hence, the speed of the car is 21 m/s whenthe driver slams on the brakes.

The car's mass does not matter because both the friction force and the car's kinetic energy are proportional to the car's mass. So, the mass gets canceled during the calculation of the speed, and speed does not appear to be a function of mass.