Speed, Time and Distance for CSAT: Trains, Boats and Circular Motion

Speed, Time and Distance is the CSAT topic that tests motion: one formula, Distance = Speed x Time, and a handful of patterns built on it. From it follows the second master idea: at fixed distance, speed and time are inversely proportional, so doubling speed halves the time. CSAT draws trains, boats in streams, circular tracks and meeting-point puzzles from this one idea, usually 2 to 4 questions a year. This article covers the core, then trains, then boats and streams, then circular motion.

The core formula and the three patterns it produces

Speed is distance per unit time. Before anything else, master the unit conversion: 1 km/h = 5/18 m/s and 1 m/s = 18/5 km/h. Nearly every CSAT error in this topic is a metre-second mix-up, not a logic error.

Worked example 1. A train runs at 90 km/h and crosses a pole in 12 seconds. Length of the train? Convert: 90 x 5/18 = 25 m/s. In 12 seconds it covers 25 x 12 = 300 metres, which is its length. A pole has no length, so the train's length equals the distance travelled while crossing it.

Worked example 2. Two cities A and B are 360 km apart. A car goes from A to B at 40 km/h and returns at 60 km/h. Average speed? The trap answer is 50, the arithmetic mean. The correct formula for equal distances is the harmonic mean: average speed = 2xy/(x + y) = 2 x 40 x 60/100 = 48 km/h (CSAT 2015). Average speed is total distance divided by total time: 720 km over 9 + 6 = 15 hours = 48 km/h.

Worked example 3. A person can walk a certain distance and drive back in 6 hours, and can walk both ways in 10 hours. Time to drive both ways? Walking one way takes 5 hours (half of 10). So driving one way takes 6 - 5 = 1 hour, and driving both ways takes 2 hours (CSAT 2013). The trick is splitting the mixed journey into its one-way legs.

Relative speed: the engine of train problems

Relative speed is the speed at which two bodies approach or separate: add the speeds when they move toward each other (or one crosses the other in opposite directions), subtract when one chases the other in the same direction. For a train crossing a person or pole, only the train's length counts; for a train crossing a platform, bridge or tunnel, the distance is train length + structure length.

Worked example 4. A 306 m train runs at 54 km/h. A person walks toward it at 3 m/s. Time to cross the person? Convert 54 km/h to 15 m/s. Opposite directions: relative speed = 15 + 3 = 18 m/s. Time = 306/18 = 17 seconds. If the person walked the same way, the relative speed would be 15 - 3 = 12 m/s and the time 25.5 seconds. Direction decides whether you add or subtract.

Worked example 5. Two trains of equal length run on parallel lines in the same direction at 66 and 30 km/h, passing each other in 50 seconds. Length of each train? Relative speed = 36 km/h = 10 m/s. In 50 seconds the faster train gains 500 m, which equals the sum of the two lengths, so each train is 250 m.

Worked example 6. A train crosses a tree in 15 seconds and a 200 m platform in 30 seconds. Length of the train? The extra 15 seconds covers the 200 m platform, so speed = 200/15 = 40/3 m/s. Train length = 15 x 40/3 = 200 m. This 'difference of times covers the structure' move solves every tree-plus-platform question.

Meeting-point problems and PYQ classics

Worked example 7. A freight train leaves Delhi for Mumbai at 40 km/h. Two hours later an express follows at 60 km/h. Where does it catch the freight? Head start = 40 x 2 = 80 km. Closing speed = 20 km/h. Catch time = 80/20 = 4 hours after the express starts, at 60 x 4 = 240 km from Delhi (CSAT 2017).

Worked example 8. A train covers 63 km at speed v, then 72 km at v + 6 km/h, taking 3 hours total. Original speed? Try the options: at 42 km/h, 63/42 = 1.5 h and 72/48 = 1.5 h, total 3 hours. Answer 42 km/h (CSAT 2013). Trying options backwards is legitimate and often fastest.

Worked example 9. Two cars start toward each other from towns 160 km apart at 8:10 am, at 50 and 30 km/h. Meeting time? Closing speed 80 km/h, time to meet = 160/80 = 2 hours, so 10:10 am (CSAT 2014).

Worked example 10. A worker is 3 minutes late walking at 5 km/h and 7 minutes early at 6 km/h. Distance? The time difference is 10 minutes = 1/6 hour. D/5 - D/6 = 1/6 gives D x 1/30 = 1/6, so D = 5 km (CSAT 2014).

Boats and streams: downstream versus upstream

In a stream, a boat's downstream speed (with the current) is boat speed + stream speed, and its upstream speed (against the current) is boat speed - stream speed. The two golden reversals: boat speed = (downstream + upstream)/2 and stream speed = (downstream - upstream)/2.

Worked example 11. In one hour a boat goes 11 km downstream and 5 km upstream. Boat speed in still water = (11 + 5)/2 = 8 km/h; stream speed = (11 - 5)/2 = 3 km/h.

Worked example 12. A man rows upstream at 8 km/h and downstream at 13 km/h. Stream speed = (13 - 8)/2 = 2.5 km/h.

Worked example 13. A man rows at 7 km/h in still water; the stream flows at 3 km/h. He rows to a place and back in 7 hours. How far is the place? Downstream 10 km/h, upstream 4 km/h. D/10 + D/4 = 7 gives 7D/20 = 7, so D = 20 km.

Worked example 14. A man rows at 9 km/h in still water; the stream flows at 4 km/h; the round trip takes 8 hours. D/13 + D/5 = 8 gives 18D/65 = 8, so D = 520/18 = 28.89 km. Same skeleton as example 13, different numbers.

Circular motion and races

Circular motion applies relative speed to a closed track. If two runners start together, the time for one to lap the other (meet after a full round) is track length / relative speed: subtract speeds for the same direction, add for opposite directions.

Worked example 15. Two athletes run on a 400 m circular track at 6 m/s and 4 m/s in the same direction. The faster laps the slower when he gains 400 m at a relative speed of 2 m/s, which takes 200 seconds. Running opposite, they meet every 400/10 = 40 seconds.

A race is the same idea on a straight line with a head start: the time for the chaser to catch up is head-start distance divided by relative speed. Worked example 16. A thief 100 m ahead runs at 8 km/h; a policeman chases at 10 km/h. Closing speed 2 km/h = 2000/3600 m/s. Catch time = 100 x 3600/2000 = 180 seconds = 3 minutes (CSAT 2013).

Traps that cost marks

First, the units trap: train lengths are in metres but speeds in km/h; convert before dividing. Second, the average-speed trap: never take the arithmetic mean of two speeds over equal distances; use 2xy/(x + y). Third, the crossing trap: a moving man or a pole contributes no length, a platform contributes its full length, and a second train contributes both trains' lengths. Fourth, the 'passes the driver of the slower train' phrasing: only the faster train's length is covered, because the driver is a point. Fifth, in boats and streams, adding when you should subtract (or vice versa) is the classic sign error: downstream is faster, upstream is slower, always.

Speed tips for the exam hall

Memorise 5/18 and 18/5 as reflexes, and pre-convert the common speeds: 36 km/h = 10 m/s, 54 km/h = 15 m/s, 72 km/h = 20 m/s, 90 km/h = 25 m/s. For any 'A to B and back' question, write average speed = 2xy/(x + y) immediately. For trains, draw the length bar in your head: train-plus-pole uses one length, train-plus-platform uses two. For meeting questions, the single formula gap/closing-speed answers almost everything. And when the equation looks messy, try the options: CSAT options are built to be testable.

Key Terms

• Speed is distance travelled per unit of time, for example 60 km/h. In CSAT it is usually given in km/h while lengths are in metres, so conversion is part of the question.
• Relative speed is the rate at which the gap between two moving bodies changes: the sum of their speeds when they move toward each other, the difference when one chases the other.
• Average speed is total distance divided by total time. Over two equal legs at speeds x and y it equals 2xy/(x + y), which is always less than the arithmetic mean.
• Downstream is motion with the current, at boat speed plus stream speed. It is always the faster leg of a boat journey.
• Upstream is motion against the current, at boat speed minus stream speed. It is always the slower leg.
• Still-water speed is the boat's own speed with no current, equal to half the sum of its downstream and upstream speeds.
• Stream speed is the current's speed, equal to half the difference of downstream and upstream speeds.
• Circular motion is movement around a closed track, where one runner laps another after gaining one full track length at the relative speed.