Kinematic

Kinematics studies motion without asking what force caused it. It connects position, Velocity, acceleration, and time. Dynamics starts when we bring in Newton’s Laws of Motion.

The derivative chain is the backbone:

$$\vec{v}(t)=\frac{d\vec{r}}{dt}$$ $$\vec{a}(t)=\frac{d\vec{v}}{dt}=\frac{d^2\vec{r}}{d{t}^2}$$

The reverse direction uses Integrals: integrate acceleration to get velocity; integrate velocity to get position.

Constant acceleration in one dimension

For simple problems we often choose one axis and assume acceleration is constant. Then the vector problem becomes scalar, as long as signs are used consistently.

Let:

• $s$ = displacement from the initial position
• $u$ = initial velocity
• $v$ = final velocity
• $a$ = constant acceleration
• $t$ = elapsed time

The standard SUVAT equations are:

Equation	Useful when
$v=u+at$	final velocity is needed
$s=ut+\frac{1}{2}a{t}^2$	displacement with time known
$s=\frac{u+v}{2}t$	average velocity is convenient
$v^2=u^2+2as$	time is not involved
$s=vt-\frac{1}{2}a{t}^2$	final velocity is known

See Equations of motion Equations and definitions for a quick formula sheet.

Sketch derivation

Start from constant acceleration:

$$a=\frac{dv}{dt}$$

Integrate from $0$ to $t$:

$$v-u=at\Rightarrow v=u+at$$

Velocity is the derivative of displacement:

$$v=\frac{ds}{dt}=u+at$$

Integrate again:

$$s=ut+\frac{1}{2}a{t}^2$$

The other SUVAT equations follow by algebraic substitution. For example, removing $t$ gives:

$$v^2=u^2+2as$$

Common use pattern

1. Draw or imagine the axis.
2. Choose the positive direction.
3. List knowns and unknowns.
4. Pick the equation missing the unwanted variable.
5. Check units and signs.

Worked examples: Equations of motion Examples. Mistakes to avoid: Equations of motion Common pitfalls.