Velocity-Time Graphs: Connecting Reaction Time, Braking, and Safe Driving

In the previous lesson, we explored how reaction time and braking distance combine to determine the total stopping distance for a car. Now, we’ll use velocity-time (V-T) graphs to analyze these ideas further and connect them to real-world driving safety, including speed limits and safe following distances.

Part 1: Understanding Velocity-Time Graphs

A velocity-time graph plots an object’s velocity (speed and direction) on the vertical axis (y-axis) against time on the horizontal axis (x-axis). These graphs help us visualize how an object’s speed changes over time and allow us to calculate important quantities like distance travelled.

Key Features of V-T Graphs

Velocity-Time Graph Example 1

Horizontal line: Constant velocity (no acceleration).
Sloped line: Changing velocity (acceleration or deceleration).
Area under the curve: Represents the total distance travelled.

Data – Analyzing Stopping with V-T Graphs

Recall the two scenarios from the previous lesson: a non-distracted driver and a distracted driver reacting to an obstacle.

Review the same video clips from P3L2. This time, focus on how the car’s velocity changes during the reaction and braking phases.

Scenario 1 - Non-Distracted Driver and Obstacle

Scenario 2 - Distracted Driver and Obstacle

Scenario 1 – Non-Distracted Driver

During reaction time, the car moves at a constant velocity (no braking yet).
During braking, the car’s velocity decreases steadily until it stops.

Scenario 2 – Distracted Driver

The reaction phase is longer, so the car travels at constant velocity for a longer time before braking begins.

In your notebook, sketch the rough shapes of velocity-time graphs for both scenarios. Label the following:

The reaction phase (constant velocity)

The braking phase (decreasing velocity)

The total stopping time

Checkpoint 1

Write the answers to the following in your notebook. It is helpful to describe regions in terms of time (i.e. $t_0$ and $t_1$)

On your V-T graph, which region represents the reaction distance?
Which region represents the braking distance?
How can you find the total stopping distance using the graph?

Part 2: Connecting Area to Distance Travelled

The area under the velocity-time graph gives the total distance travelled. For stopping scenarios: Area under a V-T graph is total distance travelled

Reaction distance: Area under the constant velocity section (rectangle).
Braking distance: Area under the sloped section (one or more triangle)
- What is the formula for the area of a triangle?.
Total stopping distance: Sum of both areas.

Use your V-T graph to estimate the reaction and braking distances for each scenario.

Compare your results to the position-time analysis from Lesson 2.

Part 3: Speed Limits, Stopping Distance, and Safe Following Distance

Why Do Speed Limits Matter?

Higher speeds mean:

Greater reaction distance (you travel farther during reaction time).
Longer braking distance (it takes more time and distance to stop).

Doubling your speed more than doubles your stopping distance!

Calculating Distance from Speed and Time

The distance traveled can be found by multiplying speed by time:

Distance = Speed × Time

For example: At 30 mph for 2 hours

$x = 30 \frac{miles}{\cancel{hour}} \times 2 \,\cancel{hour}$

$x=30 \frac{miles}{hour} \times 2 \,hour$ = 60 miles

Safe Following Distance

To avoid collisions, drivers must leave enough space to stop safely if the car in front suddenly brakes. This is called the following distance.

A common rule: “2-second rule” – Stay at least 2 seconds behind the car in front. At higher speeds or in poor conditions, increase this gap.

In your notebook, attempt these calculations to examine the “2-second rule” mathematically. The solutions are provided in the details, so be sure your notebook has accurate examples before you finish this work.

At 50 km/h (about 14 m/s or 31 mph), how far do you travel in 2 seconds?
- Solution:
  Distance = Speed × Time
  Distance = 14 m/s × 2 s = 28 meters
- Answer: You travel 28 meters in 2 seconds.
If your reaction time is 1.5 seconds and your car’s braking distance from 50 km/h is 20 meters, what is your total stopping distance?
- Solution:
  Reaction distance = Speed × Reaction time
  Reaction distance = $14 m/s \times 1.5 s$ = 21 meters Total stopping distance = Reaction distance + Braking distance
  Total stopping distance = $21 m + 20 m$ = 41 meters
- Answer: Your total stopping distance is 41 meters.
How does this compare to your following distance?
- Solution:
  Using the “2-second rule,” the safe following distance at 50 km/h is:
  Following distance = Speed × 2 seconds
  Following distance = $14 m/s × 2 s$ = 28 meters
  Compare this to the total stopping distance:
  Total stopping distance (41 meters) > Following distance (28 meters).
- Answer: Your total stopping distance is 13 meters longer than the safe following distance, meaning the “2-second rule” may not provide enough space to stop safely at this speed.

Checkpoint 2

How does increasing speed affect your reaction and braking distances?
Why is it important to adjust your following distance at higher speeds or in poor conditions to avoid accidents? Reference physics concepts in your answer.

Part 4: Visualizing Safe Driving with V-T Graphs

By analyzing velocity-time graphs, we can see how reaction time and braking combine to determine stopping distance. This helps explain why speed limits and safe following distances are critical for avoiding collisions.

In your notebook, sketch a V-T graph for a car following another at a safe distance and then stopping suddenly.

Show what happens if the following distance is too short.