Relative Speed: Add or Subtract Only After Drawing the Motion

Speed belongs to a viewpoint

Two people move along the same straight road at constant speeds. To find when they meet or one catches the other, ask how quickly the gap between them changes. That gap-changing speed is their relative speed in this one-dimensional model.

Approaching each other

Two cyclists start 300 m apart and ride towards one another at 6 m/s and 4 m/s. In one second, the first removes 6 m from the gap and the second removes another 4 m. Together the gap shrinks by 10 m each second.

Relative speed = 6 + 4 = 10 m/s. Meeting time = gap / gap-closing speed = 300/10 = 30 seconds. Check: first travels 180 m; second travels 120 m; 180 + 120 = 300 m.

Catching up in the same direction

A runner at 8 m/s is 90 m behind another runner at 5 m/s. Both continue in the same direction at constant speeds. In one second, the faster runner moves 8 m while the front runner also moves 5 m, so the gap falls by only 8 − 5 = 3 m.

Catch-up time = 90/3 = 30 seconds. Check: in 30 s the rear runner travels 240 m and the front runner 150 m. The extra 90 m exactly closes the initial gap. If the rear runner were no faster, it would not catch up under these assumptions.

Convert units before calculating

A bus at 54 km/h is 100 m behind a car at 36 km/h. Both move in the same direction at constant speeds. Relative speed = 18 km/h, but a gap stated in metres needs m/s.

1 km = 1,000 m and 1 h = 3,600 s. Thus 1 km/h = 1,000/3,600 = 5/18 m/s. Therefore 18 km/h = 18 × 5/18 = 5 m/s. A 100 m gap closes in 100/5 = 20 s.

A common mistake

“Opposite directions means add” is useful only after drawing the situation. If they move away from one another, addition gives the rate the gap grows, not a meeting time. Keep the question's gap, direction, starting positions and constant-speed assumptions visible. Dividing metres by km/h without conversion is invalid.

Check yourself

1. Approaching: 420 m gap, speeds 8 and 6 m/s. Meeting time?
2. Same direction: rear speed 7 m/s, front speed 4 m/s, initial gap 60 m. Catch-up time?
3. Convert 72 km/h to m/s.

Answers: (1) 420/(8 + 6) = 30 s. (2) 60/(7 − 4) = 20 s. (3) 72 × 5/18 = 20 m/s. All examples assume straight-line constant-speed motion.