2.5.1 Stopping distance and reaction time

Reaction time determines thinking distance

1. A reaction begins when receptors detect a stimulus, the nervous system processes it and muscles produce a response.
2. Typical measured human reaction times are about $0.2s$ to $0.9s$, but the value varies between people and conditions.
3. Thinking distance is calculated using $d=vt$, where $v$ is the vehicle's speed and $t$ is the driver's reaction time.

Stopping distance has two parts

1. The relationship is $stopping\; distance=thinking\; distance+braking\; distance$.
2. Thinking distance ends when braking begins, so the two distances are consecutive and must be added.
3. At constant reaction time, thinking distance is directly proportional to speed because $d=vt$.

Typical stopping distances

1. $20mph$: $6m$ thinking distance and $6m$ braking distance give $12m$ in total.
2. $30mph$: $9m$ thinking distance and $14m$ braking distance give $23m$ in total.
3. $40mph$: $12m$ thinking distance and $24m$ braking distance give $36m$ in total.
4. $50mph$: $15m$ thinking distance and $38m$ braking distance give $53m$ in total.
5. $60mph$: $18m$ thinking distance and $55m$ braking distance give $73m$ in total.
6. $70mph$: $21m$ thinking distance and $75m$ braking distance give $96m$ in total.
7. These values are general guides because attention, road conditions, weather, tyres and brake condition can change the actual distance.

Calculating total stopping distance

Reaction-time evidence must be evaluated

1. A mean from repeated readings is more reliable than one measurement because random variation has less effect.
2. A result for one person cannot represent all drivers because age, alertness, practice and individual differences affect reaction time.
3. A ruler-drop test measures a simple visual response and does not reproduce every decision involved in recognising a road hazard.

Distance begins before the brakes act

1. A driver with a longer reaction time travels farther before braking even when the vehicle and road are unchanged.
2. Increasing speed increases the distance travelled during every second of reaction time.

2.5.2 Factors affecting stopping distance

Stopping distance depends on driver, vehicle and road

1. Thinking distance depends on the vehicle's speed and the driver's reaction time.
2. Braking distance depends on speed, vehicle mass, brake condition, road condition and the friction between the tyres and road.
3. Any factor that increases either part increases the total stopping distance.

Speed increases both parts

1. At a fixed reaction time, thinking distance increases in direct proportion to speed because $d=vt$.
2. The kinetic energy store is $E_k=\frac{1}{2}m{v}^2$, so doubling speed gives four times the kinetic energy.
3. The brakes must transfer this extra energy, so braking distance rises much more sharply than thinking distance if braking force is similar.

Mass and braking force affect deceleration

1. A greater mass gives a greater kinetic energy store at the same speed because $E_k=\frac{1}{2}m{v}^2$.
2. For the same braking force, a greater mass also produces a smaller deceleration because $a=\frac{F}{m}$.
3. Both links mean that a more massive vehicle needs a longer braking distance under the same conditions.
4. Worn or overheated brakes can produce a smaller braking force, giving a smaller deceleration and a longer braking distance.

Road and tyres control friction

1. Water, ice, loose gravel or mud can reduce tyre-road friction, reducing the maximum braking force.
2. Worn tyres, low tread depth or unsuitable tyre pressure can reduce grip and increase braking distance.
3. A smaller braking force produces a smaller deceleration, so the vehicle travels farther before stopping.
4. Anti-lock braking systems help prevent the wheels locking, allowing the tyres to keep grip and the driver to retain steering control.

Driver factors change reaction time

1. Alcohol, illegal drugs and some medicines can slow nervous-system processing and muscle response.
2. Mobile phones, passengers, eating, loud music and adjusting controls divert attention from the road.
3. Tiredness reduces alertness and can delay hazard detection and response.
4. A longer reaction time increases thinking distance because the vehicle continues at its original speed for longer before braking.

Calculations connect energy and distance

1. If the mean braking force is treated as constant, the work done by the brakes is $W=Fs$.
2. Equating work done with the initial kinetic energy gives $Fs=\frac{1}{2}m{v}^2$, so $s=\frac{m{v}^2}{2F}$.
3. This model shows why braking distance increases with mass and with the square of speed, but decreases when braking force increases.

Every factor acts through distance or force

1. Driver factors alter reaction time and therefore thinking distance.
2. Vehicle and road factors alter kinetic energy, braking force or deceleration and therefore braking distance.

2.5.3 Dangers of large decelerations

Large decelerations produce large forces

1. A rapid stop changes velocity in a short time. The magnitude of acceleration is $|a|=\frac{v-u}{t}$, so a shorter stopping time produces a larger deceleration.
2. A large deceleration requires a large resultant force. For an object of fixed mass, $F=ma$, so the force magnitude increases in direct proportion to the deceleration magnitude.
3. Occupants continue moving when the vehicle first stops. Their inertia keeps them moving at the original velocity until a seat belt, airbag or part of the vehicle exerts a resultant force.
4. Large forces can injure the body. Injury is more likely when the force is concentrated over a small area or when different body tissues stop over different times.

Safety features reduce the force

1. Seat belts stretch slightly. This increases the time and distance over which the occupant loses momentum, reducing the average force because $F=\frac{\Delta p}{\Delta t}$.
2. Airbags compress and spread the force. Compression increases the stopping time, while the large contact area reduces pressure on the head and chest because $p=\frac{F}{A}$.
3. Crumple zones deform. Energy is transferred by work done deforming the vehicle, and the longer collision time reduces the force on the passenger compartment.
4. A rigid passenger cell limits deformation around occupants. It reduces the chance that the vehicle structure enters the space occupied by passengers.

2.5.4 Braking distance and initial speed

Braking distance depends on speed

1. Stopping distance has two parts. $stopping\; distance=thinking\; distance+braking\; distance$.
2. Thinking distance is travelled before braking begins. For a fixed reaction time, $d=vt$, so thinking distance is directly proportional to initial speed.
3. Braking distance is travelled while the brakes slow the vehicle. It depends on the vehicle's initial kinetic energy and the work done by the braking force.

Work done removes kinetic energy

1. A moving vehicle initially stores kinetic energy. $E_{\mathrm{k}}=\frac{1}{2}m{v}^2$.
2. The braking force does work against the motion. For a constant braking force, $W=Fd$, where $d$ is the braking distance.
3. Bringing the vehicle to rest transfers its initial kinetic energy. Therefore $Fd=\frac{1}{2}m{v}^2$.
4. Rearranging gives the braking-distance model. $d=\frac{m{v}^2}{2F}$.
5. For constant mass and braking force, $d\propto v^2$. Doubling speed makes braking distance four times larger, while tripling speed makes it nine times larger.
6. A larger mass increases braking distance for the same braking force. More kinetic energy must be transferred because $E_{\mathrm{k}}$ is directly proportional to mass.
7. A smaller braking force increases braking distance. Worn brakes, wet or icy roads and reduced tyre grip limit the force available, so a greater distance is required to transfer the same energy.

Estimating stopping distance

1. Convert speed to metres per second before calculating. A useful estimate is $30mph\approx 13{m\; s}^{-1}$ and $60mph\approx 27{m\; s}^{-1}$.
2. Calculate thinking distance with $d=vt$. A typical alert reaction time may be estimated as about $0.7s$ when no value is provided.
3. Calculate braking distance using energy or kinematics. Use $Fd=\frac{1}{2}m{v}^2$ when force is known, or use a valid constant-acceleration equation when deceleration is known.
4. Add the two distances only at the end. Thinking distance and braking distance respond differently to increasing speed.