Question

if you double your speed, your braking distance becomes roughly: a twice as long b about the same c four times as long d eight times as long

Explanation:

Step1: Recall kinetic - energy and braking - distance relation

The work - energy theorem is used. The work done by the brakes to stop the car is equal to the initial kinetic energy of the car, $W = \Delta K$. The work done by the brakes $W = Fd$, where $F$ is the braking force (assumed constant) and $d$ is the braking distance. The kinetic energy of the car is $K=\frac{1}{2}mv^{2}$. So, $Fd=\frac{1}{2}mv^{2}$, and $d=\frac{mv^{2}}{2F}$.

Step2: Analyze the effect of doubling the speed

Let the initial speed be $v_1 = v$ and the initial braking distance be $d_1=\frac{mv^{2}}{2F}$. If the speed is doubled, $v_2 = 2v$. Then the new braking distance $d_2=\frac{m(2v)^{2}}{2F}=\frac{4mv^{2}}{2F}$.

Step3: Find the ratio of the new and old braking distances

Calculate $\frac{d_2}{d_1}=\frac{\frac{4mv^{2}}{2F}}{\frac{mv^{2}}{2F}} = 4$.

Answer:

C. Four times as long