Chapter 6, Problem 23P

To determine

Speed of the car when the driver slammed on (and locked) the brakes.

Answer to Problem 23P

Solution:Speed of the car is $21.4\text{ m/s}$ when the driver slammed on (and locked) the brake.

Explanation of Solution

Given:Coefficient of friction, ${\mu }_k=0.3$

Force acting is normal force = mg

Formula Used:

According to work-energy theorem, work done by the force equal to change in kinetic energy

  $\begin{array}{l}W=\Delta K.E\\ W=K.E_{final}-K.E_{initial}\end{array}$

Here final velocity is zero so final kinetic energy is zero

  $F\times s=-\frac{1}{2}m{v}^2$

Because it is the (negative) work of the friction force that stops the car

Considering the friction, the above equation becomes

  ${\mu }_{k}F\times s=\frac{1}{2}m{v}^2$

Where F is the force exerted, s is the breaking distance, m is the mass of the car; v is the speed of the car, ${\mu }_k$ is coefficient of friction.

Calculations: Substitute all known values and solve for velocity v

  $\begin{array}{l}{\mu }_{k}F\times s=\frac{1}{2}m{v}^2\\ {\mu }_k\times mg\times s=\frac{1}{2}m{v}^2\\ 2{\mu }_k\times g\times s=v^2\\ v^2=2\times 0.3\times 9.81\times 78\\ v^2=459.108\\ v=21.42\text{ m/s}\end{array}$

The speed of the car is 21.42 m/s when the driver slammed on the brake.

If the accident happened on a slope then mass would not cancel out of the equation, but here mass does not matter because it took place on a level surface. Gravity acts normal to the Earth's surface. No matter what the action is, it is independent of any other force acting at $90°$ .

Conclusion:Speed of the car is $21.4\text{2 m/s}$ when the driver slammed on the brake to avoid an accident. It skids for $7\text{8}\text{m}$ long. Mass cancels out because the accident occurred on a level surface.