Kinematics of motion in a straight line

The language of motion

• How fast is a car going after 10 seconds of acceleration 加速度? How far has a ball fallen in 3 seconds?
• Kinematics 运动学 answers these questions with a small set of powerful equations — the suvat equations.

Scalars, vectors and graphs

• Distance 距离 and speed 速率 are scalars; displacement 位移, velocity 速度 and acceleration are vectors.
• On a velocity–time graph 速度-时间图: area = displacement, gradient 斜率 = acceleration.

On a velocity–time graph, the gradient is the acceleration and the area underneath is the displacement.

• Differentiate to go displacement → velocity → acceleration; integrate to go back.

The suvat equations

For constant acceleration 匀加速:

On a velocity-time graph, area is displacement and gradient is acceleration

Worked example

• From rest ($u = 0$), $a = 2.5\text{ m s}^{-2}$, $t = 4\text{ s}$:
• $v = u + at = 0 + 2.5 \times 4 = 10\text{ m s}^{-1}$.
• $s = ut + \dfrac{1}{2}at^2 = 0 + \dfrac{1}{2}(2.5)(16) = 20\text{ m}$.

Variable acceleration

• When acceleration isn't constant, use calculus:
• $v = \dfrac{ds}{dt}$, $a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2}$.
• Integrate to go the other way: $s = \int v\,dt$, $v = \int a\,dt$.
• On a velocity–time graph the area is displacement; on a displacement–time graph the gradient is velocity. Differentiation and integration move between them; a negative acceleration is deceleration.

You've got it

Write from a speed of 11 metres per second

• Initial speed 3 metres per second, acceleration 2 metres per second squared, time 4 seconds.
• Find the final speed and the displacement.