Kinematics

Distance is the total length of the actual path travelled by an object — always positive, a scalar quantity. Displacement is the shortest straight-line distance from the initial to the final position, along with direction — a vector quantity, which can be positive, negative, or zero (if the object returns to its starting point).

Speed is the rate of change of distance with time (scalar). Velocity is the rate of change of displacement with time (vector). Average speed = total distance/total time; average velocity = total displacement/total time — these can differ significantly if the path isn't a straight line.

An object is in uniform motion if it covers equal distances in equal intervals of time (constant speed). It is in non-uniform motion if it covers unequal distances in equal time intervals (changing speed) — most real-world motion, such as a car accelerating from rest, is non-uniform.

The slope of a position-time graph at any point gives the instantaneous velocity at that instant. A straight-line graph indicates constant velocity; a curved graph indicates changing velocity (acceleration); a horizontal line indicates the object is at rest.

The slope of a velocity-time graph gives instantaneous acceleration. The area under a velocity-time graph (between the curve and the time axis, over a given interval) gives the displacement during that interval.

Instantaneous velocity is the velocity at one specific instant, defined as the limiting value of average velocity as the time interval considered shrinks towards zero: $v = \lim_{\Delta t \to 0} \dfrac{\Delta x}{\Delta t} = \dfrac{dx}{dt}$. Graphically, this corresponds to the slope of the tangent drawn to the position-time curve at that particular instant.

These equations relate initial velocity ($u$), final velocity ($v$), acceleration ($a$), time ($t$), and displacement ($s$), and apply only when acceleration is constant.

On a velocity-time graph starting at initial velocity $u$ and rising steadily to final velocity $v$ over time $t$, the slope of this straight line equals acceleration: $a = \dfrac{v-u}{t}$. Rearranging directly gives $v = u + at$ — this is how the first equation of motion is derived graphically, using the definition of acceleration as the slope of a v-t graph.

The second equation follows from calculating displacement as the area under the v-t graph — this area is a trapezium, splitting into a rectangle (area $ut$) and a triangle (area $\tfrac{1}{2}t(v-u) = \tfrac{1}{2}at^2$, using $v-u=at$), giving $s = ut + \tfrac{1}{2}at^2$. The third equation is obtained by eliminating $t$ between the first two equations algebraically.

Objects falling freely under gravity alone (ignoring air resistance) experience constant acceleration $g \approx 9.8\ \text{m/s}^2$ (often rounded to $10\ \text{m/s}^2$), directed downward. All the standard equations of motion apply directly, with $a$ replaced by $g$ (taken as positive for downward motion, or negative if upward is chosen as the positive direction) — a key insight is that, ignoring air resistance, all objects fall with the same acceleration regardless of their mass.

A scalar quantity has only magnitude (e.g., mass, distance, speed, temperature). A vector quantity has both magnitude and direction (e.g., displacement, velocity, acceleration, force), and must be combined using vector rules, not simple arithmetic.

Equal vectors: same magnitude and direction. Negative vector: same magnitude but opposite direction. Unit vector: magnitude of exactly 1, used to indicate pure direction (denoted with a cap, e.g., $\hat{i}$). Null (zero) vector: zero magnitude, direction undefined/arbitrary — arises, for example, as the resultant of two equal and opposite vectors.

$\vec{A} \cdot \vec{B} = AB\cos\theta$ — a scalar result, used, for example, to calculate work done: $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$. It is maximum when the vectors are parallel ($\theta=0°$) and zero when perpendicular ($\theta=90°$).

$\vec{A} \times \vec{B} = AB\sin\theta\,\hat{n}$ — a vector result, perpendicular to both $\vec{A}$ and $\vec{B}$ (direction given by the right-hand rule), used, for example, to define torque: $\vec{\tau} = \vec{r} \times \vec{F}$. It is maximum when the vectors are perpendicular and zero when parallel.

If two vectors are represented as two sides of a triangle taken in the same order (tip-to-tail), the resultant is represented by the third side, taken in the opposite order (tail-to-tip).

Any vector can be broken down (resolved) into perpendicular components — commonly horizontal ($A\cos\theta$) and vertical ($A\sin\theta$) components, where $\theta$ is the angle the vector makes with the horizontal axis. This is extremely useful for analysing motion in 2D, like projectile motion.

The relative velocity of object A with respect to object B is the velocity that A appears to have as observed from a reference frame attached to B: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$.

A boat's velocity relative to the ground is the vector sum of its velocity relative to the water (its own rowing/engine velocity) and the water's velocity relative to the ground (the current). Correctly resolving these velocity vectors solves problems about crossing time, drift, and the direction to steer for a desired path.

A projectile is an object given an initial velocity and then allowed to move under gravity alone (ignoring air resistance). Its motion can be analysed as two independent components: uniform horizontal velocity (no horizontal acceleration) and uniformly accelerated vertical motion (acceleration $= -g$) — the horizontal and vertical motions don't affect each other.

Eliminating time from the horizontal and vertical displacement equations gives the shape of a projectile's path:

This is the equation of a parabola, confirming that projectile trajectories (under gravity alone, ignoring air resistance) are always parabolic in shape.

When an object is launched horizontally from a height (θ = 0°), its horizontal velocity remains constant throughout, while it falls under gravity exactly like an object dropped from rest — the time to reach the ground depends only on the height, not on the horizontal launch speed.

In uniform circular motion, an object moves at constant speed along a circular path, but its velocity direction is continuously changing — meaning it is still accelerating. Angular velocity ($\omega$) is the rate of change of angular position, related to linear speed by $v = \omega r$. The time period (T) is the time for one complete revolution, and frequency (f) is the number of revolutions per second, with $\omega = 2\pi f = 2\pi/T$.

Even though speed is constant, the continuously changing direction of velocity means there's a centripetal acceleration, directed towards the centre of the circle, given by $a_c = v^2/r = \omega^2 r$.

By Newton's second law, this acceleration requires a corresponding centripetal force, $F_c = ma_c = \dfrac{mv^2}{r}$, always directed towards the centre of the circular path. This isn't a new, separate type of force — it's simply the name given to whatever real force (tension, gravity, friction, normal force) happens to be providing the necessary centre-directed pull for that particular circular motion.