Time, Speed & Distance

Time-speed-distance (including trains and boats) is a heavy CDS scorer. One relation drives everything: Speed = Distance / Time. Master unit conversion and relative speed and the variants fall out.

Core relations

• Speed = Distance / Time; Distance = Speed x Time; Time = Distance / Speed.
• Convert km/h to m/s by multiplying by 5/18; m/s to km/h by multiplying by 18/5.
• Average speed = total distance / total time. For equal distances at speeds x and y, average = 2xy/(x + y) (the harmonic mean).

Relative speed

• Same direction: relative speed = difference of speeds. Opposite directions: sum of speeds.
• A train of length L crossing a pole takes L/speed; crossing a platform of length P takes (L + P)/speed.

Boats and streams

• Downstream speed = u + v; upstream speed = u - v (u = boat in still water, v = stream).
• So u = (down + up)/2 and v = (down - up)/2.

Exam Tricks & Tips

• 🎯 For equal distances, average speed = 2xy/(x+y) — never the plain average of the two speeds.
• 🎯 km/h to m/s is x5/18; the reverse is x18/5 — fix these before touching train problems.
• 🎯 Train + platform time uses (length of train + length of platform); a pole uses just the train length.
• 🎯 From boat data, u = (downstream + upstream)/2 and v = (downstream - upstream)/2.
• 🎯 If speed and distance are inversely/directly linked, use the ratio method instead of full computation.
• ❌ Common mistake: averaging two speeds arithmetically for equal distances — you must use the harmonic mean 2xy/(x+y).

Expected exam pattern

Basic speed/time, average speed, relative speed (trains) and boats-and-streams questions; 8-12 marks, formula and conversion heavy.

Quick recap

Speed = distance/time; convert with 5/18 and 18/5; average of equal distances = 2xy/(x+y); relative speed adds (opposite) or subtracts (same); use u+/-v for streams.

Cover the answer, recall, then check. 12 cards on time, speed and distance.

Q1. Convert km/h to m/s: multiply by?
A1. 5/18.

Q2. Convert m/s to km/h: multiply by?
A2. 18/5.

Q3. Average speed for equal distances at speeds x and y?
A3. 2xy/(x + y) (harmonic mean).

Q4. Relative speed of two objects moving in opposite directions?
A4. The sum of their speeds.

Q5. Relative speed of two objects moving in the same direction?
A5. The difference of their speeds.

Q6. Time for a train of length L to cross a pole?
A6. L / speed.

Q7. Time for a train (length L) to cross a platform (length P)?
A7. (L + P) / speed.

Q8. Downstream and upstream speeds in terms of u and v?
A8. Downstream = u + v; upstream = u - v.

Q9. Boat speed in still water from downstream/upstream speeds?
A9. u = (downstream + upstream)/2.

Q10. 72 km/h in m/s?
A10. 72 x 5/18 = 20 m/s.

Q11. A car covers 150 km in 2.5 hours. Average speed?
A11. 150/2.5 = 60 km/h.

Q12. Stream speed from downstream/upstream speeds?
A12. v = (downstream - upstream)/2.

Time, Speed and Distance (TSD) — including trains, boats, and relative motion — is a high-frequency CDS arithmetic topic. Everything flows from one relationship, and the tricks lie in handling relative speed and unit conversion.

Core idea / what this tests

This tests the relationship Distance = Speed × Time and its applications: average speed, relative speed (trains crossing, boats in streams), and unit conversion. The skill is choosing the right frame (relative vs absolute) for each situation.

Deep explanation

Beginner — the base relationship and units

• Distance = Speed × Time, so Speed = Distance/Time and Time = Distance/Speed.
• Unit conversion: to convert km/h to m/s, multiply by 5/18; to convert m/s to km/h, multiply by 18/5. (E.g., 72 km/h × 5/18 = 20 m/s.)

Intermediate — average and relative speed

• Average speed for equal distances at speeds a and b = 2ab/(a+b) (the harmonic mean), NOT the simple average.
• Relative speed: if two objects move in the same direction, relative speed = difference of speeds; in opposite directions, sum of speeds.
• Trains: to cross a pole, distance = train's length; to cross a platform/bridge, distance = train length + platform length.

Advanced — boats and streams

Let boat speed in still water = b and stream speed = s:

• Downstream speed = b + s (with the current); upstream speed = b − s (against it).
• From given downstream (D) and upstream (U) speeds: boat speed = (D + U)/2, stream speed = (D − U)/2.
  Similar logic applies to a plane with/against wind. For two trains crossing each other, distance = sum of both lengths, at relative speed.

Worked example

A train 150 m long crosses a platform 350 m long in 20 seconds. Find its speed in km/h.

Step 1 — total distance to cross = train length + platform length = 150 + 350 = 500 m.
Step 2 — speed = distance/time = 500 m ÷ 20 s = 25 m/s.
Step 3 — convert to km/h: 25 × 18/5 = 90 km/h.
Answer: 90 km/h. The key was adding the platform length to the train length for the crossing distance.

CDS relevance

TSD questions are numerous in CDS and reward quick unit conversion and correct relative-speed setup. Trains and boats-and-streams are especially common; the (D+U)/2 and 2ab/(a+b) formulas turn multi-step problems into single calculations, saving exam time.

Exam tricks & shortcuts

• km/h → m/s: × 5/18; m/s → km/h: × 18/5.
• Average speed (equal distances) = 2ab/(a+b), never (a+b)/2.
• Same direction ⇒ subtract speeds; opposite direction ⇒ add speeds.
• Boats: boat = (down + up)/2, stream = (down − up)/2.
• Crossing a platform ⇒ distance = train length + platform length; crossing a pole ⇒ just train length.

Key formulas

• Speed = distance/time; distance = speed × time.
• Unit conversion: 1 km/h = 5/18 m/s; 1 m/s = 18/5 km/h.
• Average speed (equal distances) = 2uv/(u+v); (equal times) = (u+v)/2.
• Same distance ⇒ time ∝ 1/speed; speeds ratio a:b ⇒ times ratio b:a.
• If speed changes to a/b of usual, time becomes b/a of usual.
• Relative speed: same direction |u−v|, opposite u+v.

Usage: convert all speeds to one unit (m/s or km/h) before combining.

Worked Example

Problem: A train 150 metres long crosses a stationary pole in 15 seconds. Find the speed of the train in km/h.

Solution:
Step 1 — Distance covered when crossing a pole. A pole has negligible length, so the train covers a distance equal to its own length:
distance = 150 m.

Step 2 — Speed in metres per second:
speed = distance / time = 150 m / 15 s = 10 m/s.

Step 3 — Convert m/s to km/h by multiplying by 18/5:
speed = 10 × 18/5 = 180/5 = 36 km/h.

Answer: The speed of the train is 36 km/h.

Solution note: To cross a pole, distance = length of train; to cross a platform, distance = length of train + length of platform.