Area Under Velocity-Time Graphs

Velocity-Time Graph Basics

A velocity-time graph shows how an object's velocity changes over time. The horizontal axis (x-axis) represents time, usually in seconds (s), and the vertical axis (y-axis) represents velocity, usually in metres per second (m/s).

The shape of the graph tells you about the motion:

• A horizontal line means constant velocity (steady speed in a straight line).
• An upward sloping line means the object is accelerating (speeding up).
• A downward sloping line means the object is decelerating (slowing down).

Velocity can be positive or negative:

• Positive velocity means the object is moving forwards.
• Negative velocity means the object is moving backwards.

This is different from speed, which is always positive. Velocity includes direction.

Area Under the Graph

The area between the velocity-time graph and the time axis represents the displacement of the object during that time interval. Displacement is a vector quantity, meaning it has both size and direction.

To find displacement, calculate the area of the shapes formed under the graph. Common shapes include:

• Rectangles: area = base × height (where height is the velocity value on the vertical axis)
• Triangles: area = ½ × base × height
• Trapeziums: area = ½ × (sum of parallel sides) × height

If the graph is above the time axis, the area is positive displacement (forward). If below, the area is negative displacement (backward).

Note that the total distance travelled is the sum of the absolute values of these areas, regardless of whether they are above or below the time axis, unlike displacement which is the net area.

For example, if a velocity-time graph shows a rectangle with a base of 4 seconds and a height of 3 m/s (above the time axis), the displacement is:

$\text{Displacement} = 4 \times 3 = 12 \text{ m}$

Interpreting Area for Motion

The total displacement is the net area under the velocity-time graph, taking into account areas above and below the time axis:

• Area above the time axis = forward displacement
• Area below the time axis = backward displacement (negative)
• Net area = total displacement (forward minus backward)

This means if an object moves forwards and then backwards, the total displacement is the difference between these two distances, not the total distance travelled.

For instance, if the area above the time axis is 15 m and the area below is 5 m, the net displacement is:

$\text{Displacement} = 15 - 5 = 10 \text{ m forwards}$