Kinematics equations of motion and graphs

The kinematics of uniformly accelerated motion in one dimension is captured by four equations linking displacement, initial velocity, final velocity, acceleration and time. Each equation omits one of these five quantities, so the choice of equation is dictated by which variable is missing.

v = u + at (no s) s = ut + ½at² (no v) s = ½(u + v)t (no a) v² = u² + 2as (no t)

Because displacement, velocity and acceleration are vectors, every calculation requires a chosen positive direction, and a negative result indicates motion in the opposite sense. The equations apply only when acceleration is constant; motion with changing acceleration must be analysed in stages or graphically.

Motion graphs provide a visual route to the same information.

• Displacement–time graphs show velocity through their gradient.
• Velocity–time graphs show acceleration through their gradient and displacement through their area.
• Acceleration–time graphs show change in velocity through their area.

For non-uniform motion, instantaneous gradients are found by drawing a tangent to the curve, and irregular areas are estimated using thin strips or by counting squares.

At a Glance

Together, the SUVAT equations and graphical methods give two complementary tools for the same physical questions, and exam questions routinely require switching fluently between them.