Speed, Distance and Time

Everything follows from distance = speed × time, applied with consistent units.

Units

To convert km/h to m/s, multiply by 5/18; to convert m/s to km/h, multiply by 18/5. So 72 km/h = 20 m/s and 54 km/h = 15 m/s.

Relative speed

When two objects move:

• in opposite directions (towards or away from each other), the relative speed is the sum of their speeds;
• in the same direction, it is the difference.

Two people 100 km apart walk towards each other at 20 and 30 km/h: they meet after 100 / (20 + 30) = 2 hours.

Trains

A train of length L must travel:

• A 200 m train at 72 km/h (20 m/s) passes a pole in 200/20 = 10 s, and a 300 m platform in 500/20 = 25 s.
• Trains of 150 m and 100 m at 60 and 30 km/h: in opposite directions, the relative speed is 90 km/h = 25 m/s, so they pass in 250/25 = 10 s; in the same direction, 30 km/h = 25/3 m/s, so 30 s.

Boats and streams

With boat speed b in still water and stream speed s: downstream speed = b + s, upstream speed = b − s. So b = (down + up)/2 and s = (down − up)/2. If a boat goes 15 km/h downstream and 9 km/h upstream, b = 12 km/h and s = 3 km/h.

kmh = lambda v: v * 5 / 18                    # km/h -> m/s

print("pole:", 200 / kmh(72), "s | platform:", (200 + 300) / kmh(72), "s")
print("trains, opposite:", round(250 / kmh(60 + 30), 2), "s | same direction:", round(250 / kmh(60 - 30), 2), "s")
down, up = 15, 9
print("boat speed:", (down + up) / 2, "km/h | stream:", (down - up) / 2, "km/h")
print("meeting time:", 100 / (20 + 30), "hours")

pole: 10.0 s | platform: 25.0 s
trains, opposite: 10.0 s | same direction: 30.0 s
boat speed: 12.0 km/h | stream: 3.0 km/h
meeting time: 2.0 hours

Practice questions