Read a physics graph before reaching for a formula
Separate the value, gradient and area of a graph, then use their units to decide what each one means.
A rising line does not always mean that an object is speeding up. A straight line does not always mean constant speed. The picture only becomes physics once you know what the axes measure. Start there, and a graph becomes a compact description of a system rather than a shape to memorise.
1. Say what each coordinate means
Read both axis labels, including their units and any scale factors. On a position–time graph, the point (4 s, 8 m) tells you where the object is at that time. On a velocity–time graph, the point (4 s, 8 m/s) tells you how fast it is moving in the positive direction. The same pair of numbers describes different information.
Check whether each axis begins at zero and whether the tick spacing is uniform. A graph is not a drawing of the route the object travelled. A downward-sloping position–time line can represent an object moving along a perfectly flat track towards smaller position values.
2. Give the gradient its own units
The gradient is the change in the vertical quantity divided by the change in the horizontal quantity. Choose two well-separated points on a straight line and calculate the changes, not just the coordinates of one point. Dividing y by x only gives the gradient when the line passes through the origin.
The units help identify the result: metres divided by seconds gives velocity; metres per second divided by seconds gives acceleration. On a curved graph, the gradient of a tangent gives the instantaneous rate of change at that point. A line joining two points gives the average rate over the interval.
3. Ask whether the area has a physical meaning
Multiplying the axis units is a useful first check. Under a velocity–time graph, (m/s) × s gives metres: the signed area is displacement. Under a force–displacement graph, N × m gives joules: the area represents work done by the force component along the displacement. Area is not automatically a useful quantity under every graph.
Keep the sign when finding displacement. A velocity of −2 m/s lasting 3 s gives a displacement of −6 m, while the distance travelled is 6 m. If velocity changes sign, add signed areas for displacement and the magnitudes of the areas for total distance.
4. Finish with a sentence about the system
Translate your calculation back into the situation: ‘The object gains 2 m/s of velocity each second’ tells you more than writing ‘gradient = 2’. Also distinguish velocity from speed. If velocity is negative and acceleration is positive, the object initially slows down as its velocity approaches zero.
Before moving on, cover your calculations and describe one interval of the graph aloud. Name the quantity that stays constant, the quantity that changes and the evidence in the graph. If the sentence disagrees with the numbers, return to the axis labels before choosing another equation.
Before you move on
Check your thinking
A horizontal line sits at +5 m/s on a velocity–time graph from 0 s to 6 s. What are the acceleration and displacement?
Show the explanation
Acceleration is 0 m/s² because the velocity does not change. Displacement is +30 m because the signed area is 5 m/s × 6 s. A horizontal velocity–time line describes constant velocity, not an object at rest.