Braking Distance

question physics short

Question

If you're driving at 30mph, see a bollard and brake. You stop just before hitting the bollard. If instead you were initially travelling at 35mph, how fast would you be moving when you hit the bollard? Attempt to estimate the answer without a calculator.


Hints

Assume that the breaks are applying a constant force in both instances. This can be solved using either suvat equations, or conservation of energy.


Answer

The problem is time-independent. Therefore, you can use the following equation: $$ v^2 = u^2 + 2as $$ The $2as$ term is constant, as the distance between the driver and the bollard, $s$, and the acceleration (braking force) are constant.

Rearranging for $2as$ and equating the two cases gives $$ \begin{aligned} v_1^2 - u_1^2 &= v_2^2 - u_2^2 \\ (30)^2 - (0)^2 &= (35)^2 - u_2^2 \\ u_2 &= \sqrt{(35)^2 - (30)^2} \end{aligned}$$

A calculator can be used to find an exact answer. However, to quickly estimate the value of $u_2$, the difference of two squares can be used: $$ \begin{aligned} \sqrt{(35)^2 - (30)^2} &= \sqrt{(35-30)(35+30)} \\ &= \sqrt{5 \cdot 75} \\ &= \sqrt{375} \end{aligned}$$

Since $19^2 < 375 < 20^2 $, the answer is approximately $19.5\text{ mph}$ .