Language of kinematics and motion graphsEdexcel A-Level Maths: Revision notes

Kinematics describes motion without considering its causes. Learn the distinctions:

• position: where an object is, relative to a fixed origin
• displacement: change in position, a vector (it has direction and sign)
• distance travelled: total length of path, always positive
• velocity: rate of change of displacement, a vector
• speed: magnitude of velocity, always positive
• acceleration: rate of change of velocity, a vector Average speed $=\frac{\text{total distance}}{\text{time}}$ and average velocity $=\frac{\text{displacement}}{\text{time}}$. If the direction reverses, distance is larger than the magnitude of displacement.

On a displacement--time graph:

• the gradient at a point is the velocity at that time
• a straight line means constant velocity
• a horizontal line means the object is at rest
• a negative gradient means motion in the negative direction
• a curve means the velocity is changing The graph shows displacement from the origin, not distance travelled. When the graph reaches its highest or lowest point, the velocity is zero as the object changes direction.

On a velocity--time graph:

• the gradient is the acceleration
• the area between the graph and the time axis is the displacement
• area below the axis counts as negative displacement
• a horizontal line is constant velocity (zero acceleration) For distance travelled, add the magnitudes of the areas above and below the axis. Split areas into rectangles and triangles, or use the trapezium area $\frac12(a+b)h$.

Worked example. A train accelerates uniformly from $0$ to $20$ m s$^{-1}$ in 10 s, then travels at $20$ m s$^{-1}$ for 30 s, then slows uniformly to rest in 20 s.

• Final acceleration: gradient $=\frac{0-20}{20}=-1$ m s$^{-2}$.
• Distance: $\frac12(10)(20)+30(20)+\frac12(20)(20)=100+600+200=900$ m.
• Average speed: $\frac{900}{60}=15$ m s$^{-1}$. Graphical solutions may be required: for example, finding when two objects have the same velocity, or the time at which a displacement is reached.