Stopping Distances

• The relationship between thinking distance and speed is linear. Because the driver's reaction time is relatively constant for a given state of mind, doubling the speed will double the thinking distance ($d = v \times t$).

• Reaction time typically ranges from 0.2 to 0.9 seconds for a healthy, alert driver. Even a small increase in this time can significantly extend the thinking distance at high speeds.

• Factors that increase thinking distance include tiredness, distractions (such as mobile phone use), and intoxication (alcohol or drugs), all of which slow down the brain's processing speed.

• The relationship between braking distance and speed is quadratic. This means that if the speed of a vehicle doubles, the braking distance increases by a factor of four ($2^2$), and if speed triples, it increases ninefold ($3^2$).

• This occurs because the work done by the brakes must equal the vehicle's initial kinetic energy. Since $KE = \frac{1}{2}mv^2$, the energy that must be dissipated is proportional to the square of the velocity.

• For a constant braking force ($F$), the work-energy principle states $F \times d = \frac{1}{2}mv^2$. Rearranging for distance gives $d = \frac{mv^2}{2F}$, confirming the $v^2$ relationship.

Comparison of Distance Components

• It is vital to distinguish between factors affecting the driver (thinking) and factors affecting the vehicle or environment (braking). For example, icy roads do not change how fast you react, but they drastically reduce the friction available for braking.

• Proportionality Checks: If a question asks how stopping distance changes when speed doubles, calculate the thinking and braking components separately. Do not simply double the total stopping distance.

• Unit Consistency: Always ensure speed is in $m/s$ and distance is in $m$ before performing calculations. If given $km/h$, divide by 3.6 first.

• Sanity Check: Remember that at high speeds, the braking distance usually becomes much larger than the thinking distance. If your calculated thinking distance is larger than your braking distance at 70 mph, re-check your math.

• Common Mistake: Students often forget to square the velocity when calculating braking distance or kinetic energy. Always write out the formula $v^2$ to avoid this error.