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Understanding Stopping & Braking Distance in Physics

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  • 0:04 Braking & Stopping
  • 1:39 Thinking Distance (TD)
  • 2:06 Braking Distance (BD)
  • 4:05 Stopping Distance (SD)
  • 5:40 Lesson Summary
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Lesson Transcript
Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

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.


Equation 1
equation1

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.


Equation 2
td

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.


Equation 3
vf

Where:

  • 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.


Diagram 1.
fbd

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.


Equation 4
a_derivation

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