Kinematics

Learning objectives

• Interpret displacement–time and velocity–time graphs.
• Choose equations for constant acceleration.
• Separate horizontal and vertical projectile motion.

The big idea

Velocity is the rate of change of displacement, and acceleration is the rate of change of velocity. Graph gradients and areas connect these quantities without requiring constant acceleration.

The familiar uniform-acceleration equations apply only while acceleration is constant. In an ideal projectile model, horizontal velocity stays constant and vertical acceleration is gravitational, with air resistance neglected.

1. Define the motion before solving it

Choose an origin, a positive direction and a time interval. Displacement is directed change of position; velocity is its rate of change. Acceleration is velocity change per second. A vehicle moving in the negative direction can have positive acceleration when it slows, so acceleration and speed do not always increase together. A sketch helps separate physical directions from algebraic signs.

2. Graphs work beyond uniform acceleration

The gradient of displacement against time gives velocity; a tangent is needed for an instantaneous value on a curve. The gradient of velocity against time gives acceleration, and signed area gives displacement. If velocity crosses zero, calculate the signed areas separately. Adding their magnitudes instead gives distance travelled, provided each interval keeps a single velocity sign.

3. Use the constant-acceleration model deliberately

Equations such as v = u + at and s = ut + ½at² require constant acceleration over the chosen interval. An object starting at 2 m/s with acceleration 3 m/s² for 4 s reaches 14 m/s and moves 32 m. Substituting an average acceleration into these equations is not generally valid when acceleration varies substantially during the interval.

4. Projectile motion has two linked parts

For an ideal projectile, resolve launch velocity horizontally and vertically. Horizontal acceleration is zero, vertical acceleration is −g if upward is positive, and the same elapsed time belongs to both equations. At the top, vertical velocity is zero but acceleration remains downward. Air resistance breaks the simple horizontal-constant-velocity model, especially for light objects moving rapidly through air.

A method you can reuse

1. Sketch axes and mark initial velocity components.
2. Decide whether acceleration is constant in each direction.
3. Select an equation containing the required unknown and known quantities.
4. Use the same time when linking projectile components.
5. Check graph areas, signs and whether the answer describes distance or displacement.

Equations and relationships

Investigate and explain

Record position at equal time intervals in a simulation, estimate velocities and compare the velocity–time area with displacement.

Compare calculated trajectories with air resistance off, then switch it on to inspect the model’s limitations. Data and graph tools support gradient and area reasoning.

Open the interactive investigation

Check your understanding