5.6.2 Calculus for Kinematics

Differentiation for Kinematics

How is differentiation used in kinematics?

Displacement, velocity and acceleration are related by calculus
In terms of differentiation and derivatives
- velocity is the rate of change of displacement
  - $v = \frac{d s}{d t}$ or $v (t) = s' (t)$
- acceleration is the rate of change of velocity
  - $a = \frac{d v}{d t}$ or $a (t) = v' (t)$
- so acceleration is also the second derivative of displacement
  - $a = \frac{d^{2} s}{d t^{2}}$ or $a (t) = s'' (t)$
If a graph is not given you can use your GDC to draw one
- you can then use your GDC’s graphing features to find gradients
  - velocity is the gradient on a displacement (-time) graph
  - acceleration is the gradient on a velocity (-time) graph

Worked Example

The displacement, $s m$ , of a particle at $t$ seconds, is modelled by $s (t) = 2 t^{3} - 27 t^{2} + 84 t$

Find $v (t)$ and $a (t)$ .
Find the times at which the particle is at rest.

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Integration for Kinematics

How is integration used in kinematics?

Since velocity is the derivative of displacement ( $v = \frac{d s}{d t}$ ) it follows that

$s = \int v d t$

Similarly, velocity will be an antiderivative of acceleration

$v = \int a d t$

How would I find the constant of integration in kinematics problems?

A boundary or initial condition would need to be known
- phrases involving the word “initial”, or “initially” are referring to time being zero, i.e. $t = 0$
- substituting the values in from the initial condition would allow the constant of integration to be found

How are definite integrals used in kinematics?

Definite integrals can be used to find the displacement of a particle between two points in time
- $\int_{t_{1}}^{t_{2}} v (t) d t$ would give the displacement of the particle between the times $t = t_{1}$ and $t = t_{2}$
- $\int_{t_{1}}^{t_{2}} |v (t)| d t$ gives the distance a particle has travelled between the times $t = t_{1}$ and $t = t_{2}$
  - Use a GDC to plot the modulus graph $y = |v (t)|$

Worked Example

A particle moving in a straight horizontal line has velocity ( $v m s^{- 2}$ ) at time $t$ seconds modelled by $v (t) = 8 t^{3} - 12 t^{2} - 2 t$ .

Given that the initial position of the particle is at the origin, find an expression for its displacement from the origin at time $t$ seconds.
Find the displacement of the particle from the origin in the first five seconds of its motion.
Find the distance travelled by the particle in the first five seconds of its motion.

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