Introduction to Velocity–time Graphs and Acceleration

In the previous chapter, Movement and position, we looked at how objects move at a steady speed. But in the real world, things rarely stay at the same speed for long. Cars speed up to overtake, and runners slow down as they get tired. This change in velocity is what we call acceleration. In this chapter, we will learn how to calculate acceleration and how to read the "story" of a journey using a velocity–time graph.

1. What is Acceleration?

Acceleration is defined as the rate at which an object changes its velocity. If you are speeding up, you are accelerating. If you are slowing down, you are still accelerating, but we often call this deceleration (or negative acceleration).

The Acceleration Formula

To calculate acceleration, you need to know how much the velocity changed and how long that change took. The formula is:

\(a = \frac{v - u}{t}\)

Where:
\(a\) = acceleration (measured in \(m/s^{2}\))
\(v\) = final velocity (measured in \(m/s\))
\(u\) = initial (starting) velocity (measured in \(m/s\))
\(t\) = time taken for the change (measured in \(s\))

Example: A cyclist accelerates from \(0 \text{ m/s}\) to \(10 \text{ m/s}\) in \(5 \text{ seconds}\).
\(a = \frac{10 - 0}{5} = 2 \text{ m/s}^{2}\).
This means every second, the cyclist's speed increases by \(2 \text{ m/s}\).

Quick Tip: Always check your units! Velocity must be in \(m/s\) and time in \(s\) for the acceleration to be in \(m/s^{2}\).

2. Understanding Velocity–time Graphs

A velocity–time (V-T) graph shows how the velocity of an object changes over a period of time. Velocity is always on the vertical axis (y-axis) and time is always on the horizontal axis (x-axis).

Reading the Shapes

The shape of the line tells you exactly what the object is doing:

  • A horizontal line: The object is moving at a constant velocity. (Note: This is different from a distance–time graph, where a horizontal line means the object has stopped!)
  • A straight line sloping upwards: The object is accelerating at a constant rate.
  • A straight line sloping downwards: The object is decelerating (slowing down) at a constant rate.
  • A line on the bottom axis (where \(v = 0\)): The object is stationary (at rest).
  • A steeper slope: This indicates a greater acceleration.

Common Mistake to Avoid: Don't confuse velocity-time graphs with distance-time graphs. Always look at the label on the y-axis before you start interpreting the graph!

3. Calculating Acceleration from a Graph

To find the acceleration from a velocity–time graph, you simply need to find the gradient (slope) of the line.

\(\text{Gradient} = \frac{\text{Change in y}}{\text{Change in x}}\)

Because the y-axis is velocity and the x-axis is time, the gradient is \(\frac{v - u}{t}\), which is our formula for acceleration.

Key Takeaway:
The gradient of a velocity–time graph = acceleration.

4. Calculating Distance from a Graph

One of the most useful things about a velocity–time graph is that it can tell you how far an object has travelled. To find the distance, you must calculate the area under the graph line.

How to do it:

If the area is a complex shape, break it down into simple geometry:

  1. Rectangles: Area = \(\text{base} \times \text{height}\)
  2. Triangles: Area = \(\frac{1}{2} \times \text{base} \times \text{height}\)

Add the areas of these shapes together to get the total distance travelled.

Did you know? Even if the graph shows the object slowing down (decelerating), the area under that line still represents the distance it covered while it was coming to a stop.

5. The Relationship between Velocity, Acceleration, and Distance

Sometimes you need to calculate a value without using a graph. There is a specific formula that links final velocity, initial velocity, acceleration, and distance (represented by the symbol \(s\)).

The Equation:
\(v^{2} = u^{2} + (2 \times a \times s)\)

Where:
\(v\) = final velocity (\(m/s\))
\(u\) = initial velocity (\(m/s\))
\(a\) = acceleration (\(m/s^{2}\))
\(s\) = distance travelled (\(m\))

Don't worry if this seems tricky at first! You will be provided with this formula on your exam equation sheet. The most important thing is to practice rearranging it to find the value you need. For example, if an object starts from rest, then \(u = 0\), which makes the calculation much simpler!

Summary: Key Points for Revision

1. Units: Always use \(m/s\) for velocity, \(s\) for time, and \(m/s^{2}\) for acceleration.
2. Acceleration Formula: \(a = \frac{v - u}{t}\).
3. V-T Graph Gradient: Finding the slope gives you the acceleration.
4. V-T Graph Area: Finding the total area under the line gives you the distance.
5. Constant Velocity: Shown as a flat horizontal line on a V-T graph.
6. The "Big" Equation: \(v^{2} = u^{2} + 2as\) is your tool for finding distance or velocity when time isn't known.

Looking for more? In the next chapter, Forces, movement and Newton's laws, we will explore what causes objects to accelerate in the first place!