Position, velocity and speed

Rest and motion are always relative. On a car ride from Kanpur to Lucknow, the passenger is at rest for the driver, but a friend left standing in Kanpur sees both of them moving. So we always say what we are measuring from.

We can treat a moving body as a point if its size is small compared with how far it moves: a car on a 300 km trip, but not a car being parked.

Two words for 'how far':

• Path length (distance): the whole length actually travelled. It has no direction (a scalar).
• Displacement: the change of position, a straight line from start to finish, with a direction (a vector).

Walk 4 m east, then 3 m north: the path length is 7 m, but the displacement is only 5 m (by Pythagoras). Go round a circle once and come back: path length 2πr, displacement zero. So path length ≥ |displacement|, and they are equal only when a body moves along a straight line without turning back.

The same split gives two kinds of 'how fast':

• Average velocity = displacement ÷ time.
• Average speed = path length ÷ time, so it is never less than the size of the average velocity.

Watch out: average speed is not the average of two speeds. Go somewhere at 40 m s⁻¹ and come back at 60 m s⁻¹: the slower trip takes longer, so the average speed is 2d ÷ (d/40 + d/60) = 48 m s⁻¹, not 50. And the average velocity is zero, because you ended where you started.

To change km h⁻¹ into m s⁻¹, multiply by 5/18 (so 72 km h⁻¹ = 20 m s⁻¹); to go back, multiply by 18/5.

The instantaneous velocity is what the average velocity becomes as the time interval shrinks to almost nothing: v = dx/dt, the slope of the tangent to the x–t graph. Over such a tiny interval a body cannot turn back, so the instantaneous speed is always the size of the instantaneous velocity. The velocity of A relative to B is v_A – v_B.