This lesson will explore the physics behind the distance it takes to stop a moving car. You'll learn the differences between thinking distance, braking distance, and stopping distance. Finally, we'll calculate how far it will take a vehicle to stop based on its initial conditions and how to estimate this distance.
Braking and Stopping
Imagine a car traveling at 73 km/hr (roughly 45 miles/hour) when the traffic light at the intersection turns from green to yellow. The driver has to make a decision - go through the yellow light or stop at the white line separating the intersection. Let's go into detail regarding the process of braking and stopping, starting with the moment the traffic light turns yellow.
The driver in our scenario decides to stop at the intersection. Yellow lights, generally speaking, do not trigger an instinctive braking response from drivers. If the driver is right at the intersection they will generally go through it. Some people drive through the intersection even after the light has turned red, which is illegal and dangerous! Our driver was at that distance from the intersection where he could come to stop safely or go through the intersection safely if he speeds up.
Making the decision to stop takes time, not a lot of time, but it does take time. The distance the car travels as a driver's brain decides when the car needs to stop until the brakes are applied is called the thinking distance (TD). The braking distance (BD) is the distance the car travels once the brakes are applied until it stops. The stopping distance (SD) is the thinking distance plus the braking distance, which is shown in Equation 1.
We can now get equations for TD and BD using kinematics and Newton's second law (ΣF = ma). This equation will only require a few variables: the reaction time of the driver (th), the car's initial speed (vo), and the coefficient of static friction between the wheels on the road.
Thinking Distance (TD)
Let's first start with the thinking distance (TD), which is shown in Equation 2. The car's velocity can be thought of being constant during the short amount of time required for the driver's reaction, so all we need is the speed times reaction time to get the thinking distance. Since the reaction time of a person wanting to brake is generally less than a second, this distance is the smallest relative to the braking distance.
Braking Distance (BD)
Deriving the equation for the braking distance is a little more involved. We start with the kinematic equation shown in Equation 3.
- vf = final velocity
- vo = initial velocity
- a = acceleration
- d = distance traveled
We know the final velocity is zero because the car has stopped. The only unknown in this equation is the acceleration a. The car decelerates (accelerating in the opposite direction of its motion) because there is an unbalanced force on it.
The brakes provide friction to the wheels slowing them down, but the static friction (f) between the wheels and the road is ultimately what stops the car. Air resistance and rolling friction are involved but to a lesser extent. The weight of the car (mg), and the normal force (N) are the vertical forces, and they are equal. A free-body diagram is shown in Diagram 1.
Newton's second law is used to calculate the acceleration of the car. Friction is calculated by multiplying the coefficient of friction (μ) and the normal force (N).
f = μ N
The normal force is mg because it only has to counter the weight of the car. The last line in Equation 4 gives us the acceleration of the car.
Now, we can plug the acceleration we just determined into Equation 3 to get the braking distance equation, BD. Let's walk through this one in a little more detail.
Equation 5. g is the acceleration due to gravity.
The last step in our derivation of the stopping distance (SD) equation is to add thinking distance (TD) to braking distance (BD), which is shown in Equation 6.
Let's pretend the reaction time for our driver is 0.5 s, and we know the initial velocity is 73 km/hr, which is 20.3 m/s. The coefficient of friction (μ) can be estimated to be 0.8, which is an average value for rubber tires on dry concrete. What can now determine our minimum stopping distance?
It is pretty amazing that, in a split second, our brains can compare the stopping distance value to our estimate of how far we are from an intersection, and make a decision whether to stop or go through the intersection. Well, this may not be exactly what happens, but with practice driving, we are trained to make good estimates of how much distance we need to stop based on our speed.
The thinking distance increases with speed. Our reaction time might be constant, but multiplying that by faster and faster speeds makes the thinking distance increase with increasing speed. Stopping distance increases exponentially with increasing speed because the initial speed of the car is squared in the braking distance equation. For example, it takes an extra 24 m to stop traveling at 20 m/s compared to 10 m/s. Graph 1 shows stopping distances compared to initial speeds.
Let's take a few moments to review what we've learned!
Whenever someone drives a car they must, at some point, bring it to a stop. This involves making the decision to stop, during which the car travels a specific distance equal to its instantaneous velocity times the driver's reaction time. We call this distance the thinking distance (TD). It is the shortest distance in the stopping distance equation because a driver's reaction time is so small.
The braking distance (BD) is the distance required to stop once the brakes have been engaged, and static friction between the tires and the road are the dominant retarding force slowing the car to a stop.
Adding these two distances together gives us the stopping distance (SD).
The largest factor in estimating this distance is the speed of the car since it is squared in the braking distance and stopping distance equations.