Time and Distance: formula sheet

Key formulas

Basic relation

Speed = Distance ÷ Time

Use consistent units. Distance = Speed × Time and Time = Distance ÷ Speed follow from it.

km/hr to m/sec

x km/hr = x × 5/18 m/sec

Use when distance is in metres or time is in seconds.

m/sec to km/hr

y m/sec = y × 18/5 km/hr

Reverse of the above.

Average speed (general)

Average speed = Total distance ÷ Total time

Works for every case.

Average speed, equal distances

Average speed = 2xy ÷ (x + y)

Only for two equal distances at speeds x and y.

Same distance, different speeds

Time ratio = inverse of speed ratio

If speeds are in ratio a : b, times are in ratio b : a.

Average speed, equal times

Average speed = (x + y) ÷ 2

Only when equal time is spent at speeds x and y.

Relative speed, opposite directions

Relative speed = S₁ + S₂

Use when bodies move towards each other or away from each other.

Relative speed, same direction

Relative speed = S₁ − S₂ (S₁ > S₂)

Use when the faster body chases or overtakes the slower one.

Time to meet or overtake

Time = Gap to be covered ÷ Relative speed

The gap is the initial distance between them, or the sum of lengths for trains crossing.

km/h to m/s

x km/h = x × 5/18 m/s

Use when distance is in metres and time in seconds.

m/s to km/h

y m/s = y × 18/5 km/h

Use when the answer must be in km/h.

Distance covered by each body at meeting

Distance₁ : Distance₂ = S₁ : S₂

Both bodies travel for the same time before meeting, so distances are in the ratio of speeds.

Basic relation

Time = Distance ÷ Speed

Keep distance and speed in matching units (metres with m/s, km with km/hr).

Train crossing a pole or standing person

Distance = Length of train

Time = length of train ÷ speed of train.

Train crossing a platform or bridge

Distance = Length of train + Length of platform

The same rule applies to bridges and tunnels.

km/hr to m/s

x km/hr = x × 5/18 m/s

Example: 72 km/hr = 72 × 5/18 = 20 m/s.

m/s to km/hr

y m/s = y × 18/5 km/hr

Example: 10 m/s = 36 km/hr.

Two trains in opposite directions

Relative speed = S1 + S2

Time to cross = (L1 + L2) ÷ (S1 + S2).

Two trains in the same direction

Relative speed = S1 − S2 (S1 > S2)

Time to cross = (L1 + L2) ÷ (S1 − S2).

Train passing a moving person

Relative speed = train speed ± person speed

Subtract if moving the same way, add if moving opposite. Distance is the train's length only.

Downstream speed

Downstream speed (x) = u + v

u = speed of boat in still water, v = speed of stream. Boat and stream move in the same direction.

Upstream speed

Upstream speed (y) = u − v

Boat moves against the stream. Needs u > v for real upstream motion.

Speed of boat in still water

u = (x + y) ÷ 2

Average of downstream and upstream speeds.

Speed of stream

v = (x − y) ÷ 2

Half the difference of downstream and upstream speeds.

Distance, speed, time

Distance = Speed × Time; Time = Distance ÷ Speed

Apply separately for each direction, using the matching speed.

Average speed for a round trip over the same distance

Average speed = 2xy ÷ (x + y)

Total distance ÷ total time. Do not take the simple average of x and y.

Distance ratio in a race

Distance of A ÷ Distance of B = Speed of A ÷ Speed of B

Valid because both run for the same time.

A beats B by d metres in a race of L metres

When A runs L, B runs (L − d). So Speed A : Speed B = L : (L − d)

B is d metres behind at the moment A finishes.

A beats B by t seconds

Distance B covers in t seconds = distance by which A beats B

Use B's speed to convert time to distance, or distance to time.

Head start

Distance run by the runner with the head start = race length − head start; the other runner runs the full race length

The head start is measured from the start line, so the runner with it runs a shorter distance.

Circular track, same direction

Time to meet = track length ÷ (faster speed − slower speed)

Applies when both start together from the same point.

Circular track, opposite directions

Time to meet = track length ÷ (sum of speeds)

Applies when both start together from the same point.

Meeting again at the starting point

Time = LCM of the times each runner takes for one lap

Use when the question asks when both are back at the start together.

Unit conversion

km/h × 5/18 = m/s

Convert before using lap length in metres.

Quick revision

• Distance = speed × time.
• km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5.
• At constant distance, speed and time are inversely proportional.
• Average speed = total distance ÷ total time.
• For equal distances at speeds x and y, average speed = 2xy ÷ (x + y).
• Opposite directions: relative speed = sum of speeds. Same direction: difference.
• A train crossing a pole or a person covers its own length.
• A train crossing a platform or another train covers the sum of the lengths.
• Downstream speed = u + v. Upstream speed = u − v.
• Still-water speed = (downstream + upstream) ÷ 2. Stream speed = (downstream − upstream) ÷ 2.
• On a circular track, bodies in opposite directions meet after track length ÷ sum of speeds.
• Always check units before calculating.

Common mistakes

• Taking the simple average of two speeds as the average speed when distances are equal Fix: Use total distance ÷ total time, or 2xy ÷ (x + y). For equal distances, the result is less than the simple average when the speeds differ.
• Multiplying by 18/5 when converting km/hr to m/sec Fix: Remember that m/sec numbers are smaller. So km/hr to m/sec uses 5/18, which reduces the number.
• Adding speeds when the bodies move in the same direction. Fix: Check the sketch first. Same direction means subtract. Opposite means add.
• Forgetting to convert km/h to m/s when lengths are in metres. Fix: Before dividing, write the units of gap and speed side by side. If they differ, convert using 5/18.
• Using only the train's length when crossing a platform. Fix: For platforms and bridges, always write distance as train length + platform length.
• Forgetting to convert km/hr to m/s. Fix: Convert speed first using × 5/18 before writing any equation.
• Adding the stream speed when going upstream, or subtracting it when going downstream. Fix: Say it in words first: with the stream means add, against the stream means subtract.
• Using the average of x and y as the average speed of a round trip. Fix: Use total distance ÷ total time, which gives 2xy ÷ (x + y) for equal distances.
• Using the beat distance as B's total distance. Fix: B runs L − d when A runs L. The beat is only the gap left.
• Adding speeds for runners moving in the same direction. Fix: Same direction means subtract speeds. Opposite direction means add.

Exam tips

• Most questions are one or two steps. Spot whether the question is a conversion, a direct S-D-T find, or an average speed question.
• If distances are equal and the options include the simple average, that option is usually a trap.
• Convert units before doing anything else. Many wrong options come from unconverted units.
• Use ratio thinking for 'speed increases by' or 'time reduces by' questions. It saves calculation time.
• Since there is no negative marking, attempt every question. Eliminate options outside the speed range and guess from the rest.
• Read the direction words carefully. "Towards each other", "opposite" and "crossing" mean add. "Overtakes", "chases" and "same direction" mean subtract.
• Check the unit of the answer asked: seconds, minutes or hours. The options are often built to catch a missed conversion.
• In train problems, ask whether the other object is a pole, a platform or another train. This decides the gap.