Position and displacement
For motion along a straight line, choose an origin and a positive direction. The position of the particle is its coordinate x.
Displacement Δx = x₂ − x₁. It can be positive, negative or zero.
Path length (distance) is the total length actually travelled, and is never negative.
Velocity
Average velocity = Δx / Δt, the displacement divided by the time interval.
Average speed = total path length / total time.
Instantaneous velocity is the velocity at a particular instant: v = dx/dt, the slope of the position–time (x–t) graph at that instant.
The magnitude of the average velocity can be less than the average speed, because displacement can be smaller than distance.
Acceleration
Acceleration a = dv/dt is the rate of change of velocity. On a velocity–time (v–t) graph:
the slope gives the acceleration,
the area under the curve gives the displacement.
The sign of acceleration does not by itself tell you whether an object is speeding up. An object speeds up when v and a have the same sign, and slows down when they have opposite signs.
Equations for uniform acceleration
For constant acceleration a, with initial velocity u at t = 0:
v = u + at
x = ut + ½at²
v² = u² + 2ax
Here x is the displacement from the starting position.
Derivation from the v–t graph. For uniform acceleration, the v–t graph is a straight line from (0, u) to (t, v). The slope gives a = (v − u)/t, which is the first equation. The area under the line is a trapezium, x = ½(u + v)t. Substituting v = u + at gives x = ut + ½at². Eliminating t between the first equation and x = ½(u + v)t gives v² = u² + 2ax.
Distance travelled in the nth second:
sₙ = u + a(2n − 1)/2
Free fall
Near the Earth's surface, ignoring air resistance, every object falls with the same constant acceleration g ≈ 9.8 m/s², directed downward. Taking upward as positive, a = −g.
Example. A ball is thrown vertically upward at 20 m/s. Take g = 10 m/s².
Time to reach the top: at the top v = 0, so 0 = 20 − 10t and t = 2 s.
Maximum height: 0 = 20² − 2 × 10 × h, so h = 20 m.
Total time of flight (back to the hand): 4 s. The ball returns at 20 m/s, now moving downward.
For an object dropped from rest, the distances fallen in successive equal time intervals are in the ratio 1 : 3 : 5 : 7 … This is Galileo's law of odd numbers.
Stopping distance
A vehicle moving at speed u that brakes with constant deceleration a comes to rest in a distance
d = u² / 2a
The stopping distance is proportional to the square of the speed. Doubling the speed makes it four times longer, which is a strong reason to obey speed limits.
Relative velocity in one dimension
If objects A and B move along the same line with velocities v_A and v_B, the velocity of A relative to B is
v_AB = v_A − v_B
Example. Two trains move in the same direction at 70 km/h and 50 km/h. The faster train's velocity relative to the slower one is 20 km/h. If they move towards each other, their relative speed is 70 + 50 = 120 km/h.