CBSE Class 10 Mathematics Time, Speed and Distance Notes Set 20

Speed, Time and Distance

Speed = distance/time, so distance = speed × time and time = distance/speed. To change km/h to m/s multiply by 5/18, and to change m/s to km/h multiply by 18/5.
Average speed = total distance/total time. For equal distances at speeds a and b the average speed is 2ab/(a + b), which is not the simple mean. If the distance is fixed, speed and time are inversely proportional.

Relative Speed and Trains

Two objects moving in opposite directions have relative speed equal to the sum of their speeds, and in the same direction the difference. A train crossing a pole or a man covers its own length; crossing a platform or a bridge it covers its length plus the length of the platform.
Two trains crossing each other cover the sum of their lengths at the relative speed. The time to catch up equals the gap divided by the relative speed.

Boats and Streams

If the boat's speed in still water is b and the speed of the stream is s, then the downstream speed is b + s and the upstream speed is b − s. Hence b = (downstream + upstream)/2 and s = (downstream − upstream)/2.
For a round trip of distance d, the total time is d/(b + s) + d/(b − s) = 2bd/(b² − s²). If a boat takes n times as long upstream as downstream, b : s = (n + 1) : (n − 1).

Races

In a race of D metres, if A beats B by d metres, then when A runs D metres B runs D − d metres. If A beats B by t seconds, then B covers the last stretch in t seconds, so B's speed = d/t. A start given to a runner reduces the distance he has to run.

Worked Examples

Illustration 1. A 150 m long train running at 54 km/h crosses a platform in 30 seconds. Find the length of the platform.
Answer:
Speed = 15 m/s, so distance in 30 s = 450 m.
Platform = 450 − 150 = 300 m.

Illustration 2. A boat goes 24 km upstream in 6 hours and 24 km downstream in 4 hours. Find the speed of the boat in still water.
Answer:
Upstream = 4 km/h and downstream = 6 km/h.
Still water speed = (4 + 6)/2 = 5 km/h.

Illustration 3. A man covers half of a distance at 40 km/h and the rest at 60 km/h. Find his average speed.
Answer:
Average speed = 2 × 40 × 60/(40 + 60) = 48 km/h.