Explaining Velocity vs Time Graphs with Common Misconceptions and a 3D Motion Simulation

A velocity vs time graph plots an object’s velocity (y-axis) against time (x-axis) to show its motion. It shows how an object’s velocity changes over time. In this post, we will learn how to explain the velocity-time graph and discuss some common misconceptions. At the end of the blog, you will find the 3D simulation of the motion using Python.

Velocity

Velocity is a vector quantity and has both magnitude and direction. It is the rate at which an object changes its position (displacement). 

On a velocity-time graph, positive velocity is represented by the line above the time axis, indicating that the object is moving forward. Negative velocity is shown when the line falls below the time axis, meaning the object is moving backward. When the line touches the time axis, the velocity is zero, and the object is momentarily at rest.

• Positive Velocity: moving forward, above the time axis
• Negative Velocity: moving backward, below the time axis
• Zero Velocity: momentarily at rest, on the time axis

Question: in the case below, when does the object change direction?

Common Mistake

• When the line changes direction (t₁ and t₃)

Correct answer

• When the line crosses over the time axis (t₂)

At t₁, the slope changes from positive (upward) to negative (downward), but the sign of the velocity does not change. The object continues to move forward and does not change direction. Similarly, at t₃, the sign of the velocity remains unchanged, and the object continues to move backward.

A change in the direction of an object can be identified by changing the sign of the velocity. If the velocity changes from positive (moving forward) to negative (moving backward), the object has reversed its direction (or vice versa). This point is represented by a line passing across the time axis from above to below (or vice versa). In this case, t₂ is the point of change in the object’s direction.

Slope

As we know, a change in velocity is called acceleration.

$$Slope=\frac{Change\;in\;Velocity}{Change\;in\;Time}=Acceleration$$

Therefore, on a velocity-time graph, the slope represents the acceleration of the object. If the line slopes upward (positive slope), velocity is increasing, so acceleration is positive. If the line is horizontal, velocity is constant, and acceleration is zero. If the line slopes downward (negative slope), velocity is decreasing, so acceleration is negative. A positive slope indicates that the object is accelerating in the positive direction, while a negative slope implies that the object is accelerating in the negative direction.

• Horizontal line: constant velocity; no acceleration
• Positive slope (upward slope): velocity is increasing; acceleration is positive
• Negative slope (downward slope): velocity is decreasing; acceleration is negative

Area under the Graph

First, let’s talk about the difference between distance and displacement. Distance refers to the total length of the path an object travels, regardless of direction (a scalar). Distance is always a positive quantity. Displacement, on the other hand, is the straight-line change in position from start to finish, including direction (a vector). Therefore, displacement can be positive, negative, or zero. So, even if you travel a long distance, your displacement can be zero if you end up where you started.

On a velocity-time graph, the area between the line and the time axis represents the displacement of the object. The area above the time axis represents positive displacement, indicating motion in the positive direction, while the area below the time axis implies negative displacement, showing motion in the opposite direction. Areas above the time axis add to the displacement, while areas below the time axis subtract from it.

Common Mistakes

When we talk about the area under a function, we mean the area bounded by the function and the time axis. The following figures give us some common mistakes in calculating displacement.

Speeding up or Slowing down

We also need to know whether the motion is speeding up or slowing down on the Velocity-Time Graph.

Common Mistake

A common mistake is to associate speeding up with an upward-sloping line and slowing down with a downward-sloping line. However, this connection only works for positive velocity.

Unlike velocity, speed is a scalar quantity and has only magnitude. In the figure below, from t₀ to t₁, the line is located below the time axis with negative velocity. We can see the line has an upward slope. However, the speed, the absolute value of velocity, is getting smaller. So the object is slowing down.

Correct Answer

Let’s look at the correct method for this problem. If the motion is further from the time axis, the object is speeding up (t₁ to t₂). If the line is moving closer to the time axis, the object is slowing down (t₀ to t₁).

• Further from the time axis – speeding up
• Closer to the time axis – slowing down

Line Shapes

On a velocity-time graph, a straight line indicates constant acceleration, or zero acceleration for a horizontal line. Straight lines show uniform motion (constant velocity) or steady speeding up/slowing down.

A curved line indicates changing acceleration (non-uniform acceleration), where the slope constantly changes. We need a tangent to find instantaneous acceleration.

3D Animation

For a better understanding of the velocity-time graph, you can check this 3D animation video.