Kinematics describes how objects move (position, velocity, acceleration).
Dynamics explains why — via forces and Newton’s laws.
For motion along a line ($x$-axis):
$$v = \frac{dx}{dt}, \quad a = \frac{dv}{dt} = \frac{d^2x}{dt^2} = v\frac{dv}{dx}$$
The relation $a = v\,\dfrac{dv}{dx}$ is useful when acceleration is given as a function of $x$.
| Equation | Variables involved |
|---|---|
| $v = u + at$ | $v, u, a, t$ |
| $s = ut + \tfrac{1}{2}at^2$ | $s, u, a, t$ |
| $v^2 = u^2 + 2as$ | $v, u, a, s$ |
| $s = \tfrac{1}{2}(u+v)t$ | $s, u, v, t$ |
where $u$ = initial velocity, $v$ = final velocity, $a$ = acceleration, $s$ = displacement, $t$ = time.
$$\mathbf{F} = m\mathbf{a}$$
In 1D: $F = ma$ (net force = mass $\times$ acceleration). Units: N = kg$\cdot$m/s$^2$.
To solve a dynamics problem:
1. Draw a free-body diagram identifying all forces.
2. Choose a positive direction.
3. Apply $F_{\text{net}} = ma$.
4. Solve for the unknown.
When force depends on time, position, or velocity:
$$m\frac{dv}{dt} = F(t, v, x) \quad \Rightarrow \text{solve the ODE}$$
Example: A 2 kg particle has force $F = 6t$ N. Find $v(t)$ given $v(0) = 0$.
$$2\frac{dv}{dt} = 6t \Rightarrow \frac{dv}{dt} = 3t \Rightarrow v = \frac{3t^2}{2} + C$$
IC: $v(0) = 0 \Rightarrow C = 0$. So $v = 1.5t^2$ m/s.
Example (resistance): A particle of mass $m$ moves with air resistance $-kv$ (opposing motion):
$$m\frac{dv}{dt} = -kv \Rightarrow v = v_0 e^{-kt/m}$$
Speed decays exponentially; particle never fully stops in finite time.
KEY TAKEAWAY: Kinematics and dynamics are linked by Newton’s second law. With a constant force, use the suvat equations; with a variable force, formulate and solve an ODE.
EXAM TIP: Write the sign convention (positive direction) explicitly before applying $F=ma$. All forces acting in the positive direction are positive; those opposing it are negative.
VCAA FOCUS: Expect questions blending calculus with kinematics — e.g., given $a(t)$ or $a(v)$, find position as a function of time or the total distance travelled.
