The following relationships link the quantities such as distance, displacement, speed, velocity, and acceleration for objects moving with constant acceleration in a straight line. With dynamics problems involving these quantities, it is useful to write out the symbols $s$, $u$, $v$, $a$, and $t$ to see what has been given in the question, what you are being asked to find, and what quantities are not relevant to the problem.

$$ d=\overline{\vphantom{}v}t $$

$$ s=\overline{\vphantom{}v}t $$

$$ v=u+at $$

$$ s=ut+\dfrac{1}{2}at^2 $$

$$ v^2=u^2+2as $$

$$ s=\dfrac{1}{2}(u+v)t $$

There are three different quantities to consider that describe motion — displacement, velocity, and acceleration. These quantities are interrelated and each of them is related to time and hence graphed along it.

Displacement-time graphs show the displacement of an object at any given time.

The gradient of a displacement-time graph relates to the speed of the object in motion.

• Where an object is moving at constant speed, the displacement of the object will change at a constant rate — the graph will show a straight line with an incline, with a gradient with the same sign as the direction of the speed; a positive gradient where the speed is in the direction defined as positive and a negative gradient where the speed is in the direction defined as negative.
• Where an object is moving with a constant acceleration, the displacement will change at a non-constant rate — the graph will show a curved line and the gradient of the curve relates to the direction of acceleration; the gradient of the curve will be increasing for acceleration in the direction defined as positive and decreasing for acceleration in the direction defined as negative.
• Where the object is stationary then the displacement will be constant — the graph will show a horizontal line.

As displacement is a vector, the direction of the object’s displacement is also considered on displacement-time graphs.

• An increasing positive displacement means that the object is getting farther away from the initial point of measurement [rest displacement or where the displacement equals zero] — travelling in the direction defined as positive, with the object’s physical displacement also in the positive direction.
• A decreasing positive displacement means that the object is getting closer to the initial point of measurement — travelling in the direction defined as negative, but with the object’s physical displacement in the positive direction.