Reading motion graphs

On a position-time graph a flat line means standing still, and a steeper line means faster.

Position-time graphs: the slope is the velocity

A graph turns a whole trip into one picture. On a position-time graph, time runs along the bottom and position goes up the side. A walker moving at a steady 2 m/s makes a straight line that rises 2 m for every second. The steepness of that line, its slope, is the velocity: slope = rise ÷ run = Δx ÷ Δt. A steeper line means a faster walker. A flat line means the position is not changing, so the walker is standing still. A line that slopes downward means the position is getting smaller, so the walker is heading back towards the reference point with a negative velocity.

When the line on a position-time graph curves, the velocity is changing. A curve that gets steeper and steeper is speeding up. A ball dropped from a window makes such a curve, a half parabola, because its distance grows with the square of time. To find the velocity at one instant on a curve, draw a straight line that just touches the curve at that point, a tangent, and measure its slope.

Velocity-time graphs: slope is acceleration, area is displacement

A velocity-time graph has velocity up the side. Here the slope tells you the acceleration: a line rising 3 m/s each second means a = 3 m/s². A flat line means constant velocity and zero acceleration. A line going down means the object is slowing down if the velocity is positive.

There is a second thing hidden in a velocity-time graph: the area between the line and the time axis equals the displacement. For constant velocity this is plain multiplication, because the area is a rectangle: 10 m/s for 5 s gives a rectangle of 10 × 5 = 50 m. For steady acceleration from rest the area is a triangle, ½ × base × height = ½ × t × (a t) = ½ a t², which is the same formula as in the last lesson. The graph gives you the equation for free.

A worked trip

A bus pulls away from a stop, speeding up steadily to 12 m/s in 6 s. It cruises at 12 m/s for 20 s, then brakes steadily to rest in 4 s. On a velocity-time graph this is a trapezium. The three areas are ½ × 6 × 12 = 36 m, 20 × 12 = 240 m and ½ × 4 × 12 = 24 m, so the bus travelled 300 m between stops. Its average speed was 300 ÷ 30 = 10 m/s, a little less than its cruising speed because of the start and the stop.

When you read any motion graph, first check the labels on both axes. The same shape means something completely different on a position graph and on a velocity graph: a flat line is "standing still" on one and "moving at steady speed" on the other.

Graphs also make it easy to compare two movers. Draw a walker at 1.5 m/s and a cyclist at 5 m/s on the same position-time axes, both starting from the same point. The cyclist’s line is steeper, and the vertical gap between the two lines at any time is how far apart they are. If the walker had a head start, their line would begin higher up the axis, and the moment the lines cross is the moment the cyclist catches up.