Acceleration equations and velocity–time graphs

Take the direction of travel as positive. When an object slows down without changing direction, its final velocity is smaller than its initial velocity, so $v-u$ is negative. Its acceleration is therefore negative; this slowing down is called deceleration.

Reading a velocity–time graph

A velocity–time graph shows time on the horizontal axis and velocity on the vertical axis. Each point tells you the object's velocity at that time. The following graph shows straight-line motion.

Data for Acceleration, constant velocity and deceleration

From 0 to 4 seconds, velocity rises steadily from 0 to $8\text{ m/s}$. From 4 to 7 seconds, velocity stays at $8\text{ m/s}$. From 7 to 9 seconds, the object slows steadily to rest.

The gradient gives acceleration:

• A straight rising line shows constant positive acceleration.
• A horizontal line shows zero acceleration: velocity is constant. It means the object is stationary only if the velocity is zero.
• A straight falling line shows constant negative acceleration.

On the same axes, a steeper slope means a greater magnitude of acceleration. The falling section here is steeper than the rising section, so the object slows down more rapidly than it initially speeds up. A curved line has a changing gradient, showing changing acceleration.

Calculating acceleration from a straight section

Choose two points on the same straight section. A gradient triangle shows the change in velocity vertically and the corresponding change in time horizontally:

$$a = \frac{\text{change in velocity}}{\text{change in time}}$$

For the rising section of the illustration:

$$a = \frac{8-0}{4-0} = 2\text{ m/s}^2$$

For the falling section:

$$a = \frac{0-8}{9-7} = -4\text{ m/s}^2$$

The negative sign describes the direction of the acceleration. Its magnitude is $4\text{ m/s}^2$: the velocity decreases by $4\text{ m/s}$ each second. These gradient calculations apply to straight sections, where acceleration is uniform.

Finding distance from the area

For motion in one direction with non-negative velocity, the area between a velocity–time graph and the time axis gives the distance travelled. A rectangle represents constant velocity: its width is time and its height is velocity, so its area is velocity multiplied by time.

For straight sections showing uniform acceleration or deceleration, split the area into rectangles and triangles:

$$\text{rectangle area} = \text{base}\times\text{height}$$ $$\text{triangle area} = \frac{1}{2}\times\text{base}\times\text{height}$$

In the illustration, the three sections give:

• First triangle: $\frac{1}{2}\times4\times8 = 16\text{ m}$.
• Rectangle: $(7-4)\times8 = 24\text{ m}$.
• Final triangle: $\frac{1}{2}\times(9-7)\times8 = 8\text{ m}$.

The total distance is $16+24+8 = 48\text{ m}$. The units explain why graph area represents distance: seconds multiplied by metres per second gives metres. A section that starts above zero velocity can be split into a rectangle with a triangle above it.

Linking velocity, acceleration and distance without time

For uniform acceleration, another useful equation is:

$$v^2-u^2 = 2ax$$

Here, $x$ is the distance travelled in metres; $u$, $v$ and $a$ have the meanings and units used earlier. This equation is useful when time is not given or needed.

For example, a car accelerates steadily from rest at $2.5\text{ m/s}^2$ until it reaches $16\text{ m/s}$. Starting from rest means $u=0$. Rearranging to find distance:

$$x = \frac{v^2-u^2}{2a}$$ $$x = \frac{16^2-0^2}{2\times2.5} = 51.2\text{ m}$$

The car travels 51.2 metres while gaining this speed. The same equation can be rearranged to find acceleration, $a=(v^2-u^2)/(2x)$, or final velocity. For final velocity, first calculate $v^2=u^2+2ax$, then take the square root, choosing the positive value for motion in the positive direction.

$x$ is distance in metres. Use only for uniform acceleration.

$$x = \frac{v^2-u^2}{2a}, \qquad a = \frac{v^2-u^2}{2x}$$

Velocity–time graphs

• Gradient = acceleration: change in velocity ÷ change in time.
• Straight slope: uniform acceleration; steeper slope: greater acceleration magnitude.
• Horizontal line: constant velocity, zero acceleration.
• Curved line: changing acceleration.
• Area between line and time axis = distance for non-negative velocity. For straight sections, split into rectangles and triangles and add their areas.

Rectangle: base × height. Triangle: ½ × base × height. Use time intervals for bases and velocities for heights.