Time, Speed & Distance

Speed says how much distance is covered in one unit of time. Distance = speed × time when speed stays constant.

For relative motion, use the sum of speeds when closing in opposite directions and the difference when catching up. Convert km/h to m/s by multiplying by 5/18.

A cyclist travels at 12 kilometres per hour for 2 hours. How far does she go? 1. Keep the units consistent: kilometres per hour and hours. 2. Multiply speed by time: 12 × 2. 3. The distance is 24 kilometres.

Each one-hour interval adds 12 kilometres because the speed is constant. The final distance is therefore 24 kilometres. If speed changes, calculate each segment separately and add the distances. An average speed must use total distance divided by total time, rather than automatically averaging two speeds.

A 180 m train overtakes a 120 m train travelling in the same direction at 54 km/h. The faster train travels at 72 km/h. Relative speed=(72-54) x 5/18=5 m/s. To pass completely it must gain 180+120=300 m, taking 60 seconds. A stationary observer would instead use only the passing train's length and actual speed.

Convert minutes to hours before using a speed given in kilometres per hour. Identify what must be fully crossed. A platform and another train both add length; a point observer does not.

Do not take the simple average of two speeds unless the time spent at each speed is equal.

A train clears a pole after moving its own length. To clear a platform, it moves the sum of train and platform lengths.

A 120 m train crosses a 180 m platform at 54 km/h. Convert speed to 15 m/s; distance is 300 m, so crossing time is 20 seconds.

Downstream speed = still-water speed + stream speed. Upstream speed subtracts the stream, assuming the same effort in both directions.

Downstream and upstream speeds are 15 and 9 km/h. Adding them gives twice the still-water speed, so it is 12; stream speed is (15-9)/2 = 3.

For relative motion, use the sum of speeds when closing in opposite directions and the difference when catching up. Convert km/h to m/s by multiplying by 5/18.

A 180 m train overtakes a 120 m train travelling in the same direction at 54 km/h. The faster train travels at 72 km/h. Relative speed=(72-54) x 5/18=5 m/s. To pass completely it must gain 180+120=300 m, taking 60 seconds. A stationary observer would instead use only the passing train's length and actual speed.

Identify what must be fully crossed. A platform and another train both add length; a point observer does not.

Distance = speed × time. Use matching units: multiply km/h by 5/18 to obtain m/s, and multiply m/s by 18/5 to obtain km/h. Average speed is total distance divided by total elapsed time, including stops when the question includes them.

For equal distances at positive speeds u and v, average speed = 2uv/(u+v). For equal times it is (u+v)/2. Relative speed is |u−v| in the same direction and u+v in opposite directions. A train clearing a pole travels its own length; clearing a platform or another train requires the sum of the lengths. For a boat with still-water speed b and stream speed r, downstream speed = b+r and upstream speed = b−r; an upstream journey requires b>r. Thus b=(downstream+upstream)/2 and r=(downstream−upstream)/2.

A 120 m train passes a 180 m platform at 54 km/h. Speed = 54 × 5/18 = 15 m/s. It travels 300 m to clear the platform, taking 300/15 = 20 seconds. For equal 60 km legs at 30 and 60 km/h, average speed = 120/(2+1) = 40 km/h. For two runners starting together on a circular track of circumference C, their next meeting occurs after C/|u−v| in the same direction when u≠v, or C/(u+v) in opposite directions. Returning together to the start is a different condition: use a common multiple of their lap times when those times are commensurable. In a 100 m race where A beats B by 20 m, their speed ratio is 100:80 under constant-speed assumptions.

Do not average speeds without checking time or distance weights. A head start changes a runner's distance or starting time; translate the stated race conditions before writing an equation.