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A graph of motion carries more information than a list of values. Two operations extract it: the gives a rate of change, and the underneath gives an accumulated total. Everything in kinematics follows from those two.
The big picture
It is worth noticing early that gradient and area are differentiation and integration in disguise. For straight-line graphs you can find them with triangles and rectangles; for curves you need calculus, which is exactly what the variable-acceleration lesson introduces. So this lesson is not a preliminary to be got through — it is the geometric picture that makes the calculus later feel inevitable rather than arbitrary. A student who can read a velocity–time graph fluently already understands what means before meeting the notation.
What you'll be able to do
On a displacement–time graph the gradient is the . A straight line means constant velocity; a horizontal line means the object is at rest; a steeper line means faster motion.
A negative gradient means motion in the negative direction — back towards, and possibly past, the starting point. Where the graph crosses the time axis, the displacement is zero and the object has returned to its start.
For a curve, the velocity at an instant is the gradient of the at that point. A curve bending upwards means the velocity is increasing.
Tip — The area under a displacement–time graph has no physical meaning. Only velocity–time and acceleration–time graphs have useful areas.
On a velocity–time graph the gradient is the and the area between the graph and the time axis is the .
A horizontal line means constant velocity and therefore zero acceleration. A straight sloping line means constant acceleration — which is precisely the situation the suvat equations describe.
Area below the axis counts as negative displacement, because the object is moving backwards during that interval. This is where distance and displacement separate: displacement is the signed total, distance is the sum of the magnitudes.
Splitting the area into triangles and rectangles works for any straight-line graph and keeps every step visible for method marks. It is also more reliable than trying to apply a trapezium formula to a shape with several sections.
When a velocity–time graph crosses the time axis, the object reverses direction. The area above the axis is positive displacement and the area below is negative.
For , add the signed areas — the negative region partially or wholly cancels the positive one. For , add the magnitudes, treating every area as positive.
Questions distinguish these deliberately, and the difference is often the whole point of including a region below the axis.
Tip — If a question asks for "the distance travelled" and the graph dips below the axis, the answer is not the same as the displacement. Check for a crossing before answering.
A curved velocity–time graph means the acceleration is changing. The acceleration at an instant is the gradient of the tangent at that point, found by drawing the tangent and computing its gradient.
The area under a curve cannot be split into exact triangles, so it is estimated — by counting squares, or by using the trapezium rule. Both give an approximation, and questions usually say which to use.
A curve that is getting steeper means the acceleration is increasing; one that is flattening means the acceleration is decreasing, even while the velocity itself continues to rise.
That description is exactly terminal velocity, and it shows why reading gradients matters: "velocity increasing" and "acceleration decreasing" are simultaneously true and not contradictory.
Think like an examiner
Common misconceptions
Reading motion graphs
Stretch yourself
A particle starts at rest and accelerates uniformly at for 5 s. It then decelerates uniformly, coming to rest after a further 5 s, and continues to accelerate in the negative direction at the same rate for 5 s more. Sketch the velocity–time graph and find the displacement and total distance.
Hint — Work out the velocity at each key time first, then compute the areas separately, watching signs.
Questions students ask
Key takeaways
How this fits the course
Related
Leads to
Test yourself
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