Velocity-Time Graphs: Understanding Motion Through Graphs

Introduction to Velocity-Time Graphs

Velocity-time (v-t) graphs are one of the most powerful tools in kinematics. Unlike position-time graphs that show where an object is, v-t graphs show how fast an object is moving and in which direction at every moment in time.

What makes v-t graphs special is that they encode TWO pieces of information simultaneously:

• The slope tells you the acceleration - how quickly velocity is changing
• The area under the curve tells you the displacement - how far the object has traveled

This interactive simulation lets you draw your own v-t graphs and immediately see how an object would move according to that graph. You'll develop an intuitive understanding of the relationship between velocity, acceleration, and displacement.

Key Concepts

1. Slope = Acceleration

The slope of a v-t graph at any point gives the instantaneous acceleration:

$a = \frac{\Delta v}{\Delta t} = \frac{v_2 - v_1}{t_2 - t_1}$

2. Area Under Curve = Displacement

The area between the v-t curve and the time axis represents the total displacement:

$\Delta x = \int_{t_1}^{t_2} v \, dt$

For simple shapes:

• Rectangle: Area = base x height = v x t
• Triangle: Area = 1/2 x base x height = 1/2 x t x v
• Trapezoid: Area = 1/2 x (v1 + v2) x t

3. Sign Conventions

Learning Objectives

After completing this lesson, you will be able to:

1. Interpret velocity-time graphs to describe motion qualitatively
2. Calculate acceleration from the slope of a v-t graph
3. Determine displacement from the area under a v-t curve
4. Predict position-time graphs from velocity-time graphs
5. Distinguish between positive and negative velocities
6. Understand how v-t graphs relate to actual object motion

Guided Exploration Activities

Activity 1: Constant Velocity

Goal: Understand what uniform motion looks like on a v-t graph

Steps:

1. Select the "Constant Velocity" preset
2. Press Play and observe the object motion
3. Notice: The v-t graph is a horizontal line at v = 10 m/s
4. Notice: The p-t graph is a straight line with constant slope
5. Calculate: What is the displacement after 10 seconds? (Use area!)

Checkpoint: Why is the acceleration zero for constant velocity?

Activity 2: Constant Acceleration

Goal: See how uniform acceleration appears on v-t and p-t graphs

Steps:

1. Select the "Constant Acceleration" preset
2. Observe the straight diagonal line on the v-t graph
3. Watch how velocity increases linearly with time
4. Notice: The p-t graph is a parabola (curved)
5. Calculate the acceleration from the slope: a = (20 - 0) / (10 - 0) = 2 m/s^2

Checkpoint: Why is the p-t graph curved while the v-t graph is straight?

Activity 3: Deceleration to Stop

Goal: Understand braking motion

Steps:

1. Select "Deceleration to Stop" preset
2. The object starts fast (15 m/s) and slows to a stop
3. Calculate: What is the deceleration? (Note: it's negative acceleration)
4. Measure the total displacement from the triangular area
5. Verify: Area = 1/2 x 10s x 15 m/s = 75 m

Activity 4: Back and Forth Motion

Goal: Interpret motion that changes direction

Steps:

1. Select "Back and Forth" preset
2. Watch the v-t graph cross the time axis (velocity becomes negative)
3. The object moves forward, stops, then moves backward
4. Notice where the p-t graph reaches its maximum (when v = 0)
5. Calculate total displacement vs. total distance traveled

Discussion: What's the difference between displacement and distance?

Activity 5: Custom Graph Drawing

Goal: Create your own motion scenario

Steps:

1. Select "Custom (Draw)" preset
2. Click and drag points to create a unique v-t graph
3. Try to create:
   • Motion that returns to the starting point
   • Maximum displacement of exactly 50 m
   • An object that stops at t = 5s then reverses

Common Mistakes to Avoid

1. Confusing slope with value - The slope gives acceleration, not velocity. The height gives velocity.

2. Forgetting negative areas - Area below the time axis represents negative displacement (moving backward).

3. Mixing up graphs - On a v-t graph, a horizontal line means constant velocity. On a p-t graph, a horizontal line means the object is stationary.

4. Ignoring units - Slope = Deltavy/Deltax has units of (m/s)/s = m/s^2. Area = v x t has units of m/s x s = m.

5. Assuming constant acceleration - If the v-t line is curved, acceleration is changing!

Challenge Questions

1. (Easy) A v-t graph shows a horizontal line at v = 8 m/s for 5 seconds. What is the displacement?

2. (Easy) An object's v-t graph shows velocity changing from 0 to 20 m/s in 4 seconds. What is the acceleration?

3. (Medium) A v-t graph is a triangle: velocity starts at 0, increases to 12 m/s at t = 3s, then returns to 0 at t = 6s. What is the total displacement?

4. (Medium) An object's v-t graph shows velocity of +5 m/s for 4 seconds, then -5 m/s for 4 seconds. What is the final displacement?

5. (Hard) Create a v-t graph where the object ends up 30 m to the left of where it started but travels a total distance of 50 m.

FAQ

Q: Can an object have positive velocity but negative acceleration? A: Yes! This means the object is moving forward but slowing down (decelerating). For example, a car moving forward while braking.

Q: What does it mean if the v-t graph crosses the time axis? A: The object momentarily stops (v = 0) and then reverses direction. The velocity changes from positive to negative or vice versa.

Q: How do I find average velocity from a v-t graph? A: Calculate the total displacement (net area under curve) and divide by total time. This is NOT the same as averaging the velocities at each instant.

Q: Why does constant acceleration produce a curved p-t graph? A: Because position changes by larger and larger amounts as velocity increases. If v = at, then x = 1/2 at^2, which is a parabola. [1]

Q: What's the difference between speed and velocity? A: Velocity includes direction (can be negative), while speed is always positive. On a v-t graph, speeds below the axis represent negative velocities.

Real-World Applications

References

1. Knight, R.D. (2017). Physics for Scientists and Engineers, 4th ed. Pearson. Chapter 2: Kinematics in One Dimension.

2. Halliday, D., Resnick, R., & Walker, J. (2018). Fundamentals of Physics, 11th ed. Wiley. Chapter 2: Motion Along a Straight Line.

3. Alberta Education. (2014). Physics 20-30 Program of Studies. https://education.alberta.ca/physics-20-30/programs-of-study/

4. OpenStax. (2021). College Physics 2e. Chapter 2: Kinematics. https://openstax.org/books/college-physics-2e/pages/2-introduction-to-one-dimensional-kinematics

5. The Physics Classroom. "Meaning of Shape for a v-t Graph." https://www.physicsclassroom.com/class/1DKin/Lesson-4/Meaning-of-Shape-for-a-v-t-Graph

6. Khan Academy. "Velocity-time graphs." https://www.khanacademy.org/science/physics/one-dimensional-motion/kinematic-formulas/v/velocity-time-graphs

7. NIST Reference on SI Units. https://www.nist.gov/pml/weights-and-measures/metric-si/si-units

About the Data

This simulation uses numerical integration to compute position from velocity:

• Time range: 0-10 seconds
• Velocity range: -20 to +20 m/s
• Number of control points: 21 (one every 0.5 seconds)
• Integration method: Trapezoidal rule
• Animation rate: 50 ms per 0.05 s simulation time

The position-time graph is computed in real-time using: $x_{i+1} = x_i + \frac{v_i + v_{i+1}}{2} \cdot \Delta t$

Verification Log

How to Cite This Simulation

APA Format:

Simulations4All. (2025). Velocity-Time Graphs Interactive Lesson [Interactive simulation]. https://simulations4all.com/simulations/physics20-velocity-time-graphs-lesson

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Keywords: velocity-time graph, v-t graph, kinematics, acceleration from slope, displacement from area, motion graphs, Alberta Physics 20, physics simulation