# Comparison of Stopping Distance and Time for Two Vehicles

## Comparison of Stopping Distance and Time for Two Vehicles

When the body is moving with a certain velocity and suddenly brakes are applied, then the body stops completely after covering a certain distance, this is called as Stopping Distance. It is the distance travelled between the time when the body decides to stop a moving vehicle and the time when the vehicle stops completely. Two vehicles of masses $${{m}_{1}}$$ and $${{m}_{2}}$$ are moving with velocities $${{v}_{1}}$$ and$${{v}_{2}}$$, when they are stopped by the same reading force (F).

The ratio of their stopping distances, $$\frac{{{x}_{1}}}{{{x}_{2}}}=\frac{{{E}_{1}}}{{{E}_{2}}}=\frac{{{m}_{1}}v_{1}^{2}}{{{m}_{2}}v_{2}^{2}}$$,

The ratio of their stopping time, $$\frac{{{t}_{1}}}{{{t}_{2}}}=\frac{{{P}_{1}}}{{{P}_{2}}}=\frac{{{m}_{1}}{{v}_{1}}}{{{m}_{2}}{{v}_{2}}}$$,

(i) If vehicles possess same velocities, $${{v}_{1}}={{v}_{2}}$$, $$\frac{{{x}{1}}}{{{x}{2}}}=\frac{{{m}{1}}}{{{m}{2}}}$$;  $$\frac{{{t}_{1}}}{{{t}_{2}}}=\frac{{{m}_{1}}}{{{m}_{2}}}$$,

(ii) If vehicles possess same kinetic momentum, $${{P}_{1}}={{P}_{2}}$$,

$$\frac{{{x}_{1}}}{{{x}_{2}}}=\frac{{{E}_{1}}}{{{E}_{2}}}=\left( \frac{P_{1}^{2}}{2{{m}_{1}}} \right)\left( \frac{2{{m}_{2}}}{P_{2}^{2}} \right)=\frac{{{m}_{2}}}{{{m}_{1}}}$$ $$\Rightarrow \frac{{{t}_{1}}}{{{t}_{2}}}=\frac{{{P}_{1}}}{{{P}_{2}}}=1$$,

(iii) If vehicle possess same kinetic energy:

$$\frac{{{x}_{1}}}{{{x}_{2}}}=\frac{{{E}_{1}}}{{{E}_{2}}}=1$$  $$\Rightarrow \frac{{{t}_{1}}}{{{t}_{2}}}=\frac{{{P}_{1}}}{{{P}_{2}}}=\frac{\sqrt{2{{m}_{1}}{{E}_{1}}}}{\sqrt{2{{m}_{2}}{{E}_{2}}}}=\sqrt{\frac{{{m}_{1}}}{{{m}_{2}}}}$$,

If vehicle is stopped by friction then,

Stopping distance $$\left( x \right)=\frac{\frac{1}{2}m{{v}^{2}}}{F}=\frac{\frac{1}{2}m{{v}^{2}}}{ma}=\frac{{{v}^{2}}}{2\mu g}$$ $$\left[ as,\,\,a=\mu g \right]$$,

Stopping time $$\left( t \right)=\frac{mv}{F}=\frac{mv}{m\mu g}=\frac{v}{\mu g}$$.