Speed, Distance & Time

Speed is ground covered divided by time taken, $v = d/t$, and it is worth being clear about what kind of statement that is. It is not a law of nature that could turn out to be false — it is the definition of average speed. Nothing in it can be wrong; it can only be misapplied. What it gives you is the single steady speed that would have covered the same distance in the same time, which is a genuinely useful summary of a trip and tells you almost nothing about any particular moment within it.

Take a drive of 148 km that takes 1 hour 45 minutes. Convert the time to a single unit first — 1.75 h — and $v = 148/1.75 = 84.6$ km/h. In SI the same trip is 148 000 m over 6300 s, giving 23.5 m/s. Both answers describe the same drive, and the traffic light you sat at for ninety seconds is buried inside both of them.

This is the zeroth member of the kinematics family: set $a = 0$ in $d = v_0 t + \tfrac{1}{2}at^2$ and you are left with $d = vt$. Run in the other direction, toward instantaneous speed, it becomes the derivative $v = \mathrm{d}d/\mathrm{d}t$, which is what your speedometer reads and what calculus was partly invented to handle. The relation also underwrites the modern definition of length itself: since 1983 the metre has been defined by fixing the speed of light at 299 792 458 m/s exactly, so distance is now measured by timing light and rearranging this formula for $d$.

The classic mistake is averaging the speeds instead of the trip. Drive 60 km out at 60 km/h and return at 30 km/h, and the average for the round trip is not 45 km/h. The outbound leg takes 1 h, the return 2 h, so it is 120 km in 3 h — 40 km/h. Average speed is always total distance over total time, never the mean of the individual speeds, because you spend longer at the slow one. The second trap is units: this formula does not convert anything, so 100 km and 30 minutes gives 3.33 in units of km/min, not km/h. Fix the units before the arithmetic, or use the unit selectors on this page. And note that this is distance, the ground covered, not displacement — a runner who finishes a 400 m lap where she started has run at a respectable speed and has an average velocity of exactly zero.