Basic Formulas
Speed equals distance over time. To convert kilometres per hour to metres per second, multiply by 5 over 18. To convert metres per second to kilometres per hour, multiply by 18 over 5. So 36 kilometres per hour is 10 metres per second and 72 is 20. Average speed is total distance over total time. When the same distance is covered at speeds x and y, the average speed is 2xy over x plus y.
Relative Speed
Objects moving in the same direction have relative speed equal to the difference of their speeds. Objects moving toward each other have relative speed equal to the sum. Meeting time equals the distance between them over the sum of speeds. Catch-up time equals the head start distance over the difference of speeds.
Trains
A train crossing a pole or person takes its length divided by its speed. Crossing a platform or bridge takes the sum of the two lengths divided by the speed. Two trains moving in opposite directions cross in the sum of their lengths divided by the sum of their speeds. In the same direction they cross in the sum of lengths divided by the difference of speeds.
Boats and Streams
Let the boat speed in still water be u and the stream speed be v. Downstream speed is u plus v and upstream speed is u minus v. The boat speed is half the sum of downstream and upstream speeds, and the stream speed is half their difference.
Shortcuts
Convert everything to the same unit first. Never use the simple average of two speeds for equal distances. Remember to add the train's own length.
Worked Examples
Example 1: Convert 72 kilometres per hour to metres per second.
Solution: 72 times 5 over 18 equals 20.
Example 2: A 150 metre train at 54 kilometres per hour crosses a pole. How long?
Solution: The speed is 15 metres per second, so the time is 10 seconds.
Example 3: The downstream speed is 15 and the upstream speed is 9. Find the boat and stream speeds.
Solution: The boat speed is 12 and the stream speed is 3.
Quick Check
Question 1: What is the average speed for equal distances at speeds x and y?
Answer: 2xy over x plus y.
Question 2: How do you convert kilometres per hour to metres per second?
Answer: Multiply by 5 over 18.
Question 3: What is the downstream speed of a boat?
Answer: Boat speed plus stream speed.