Time, speed and distance

In short

Time, speed and distance (TSD) studies uniform motion: the relation speed = distance ÷ time and everything built on it — average speed, relative speed, meeting and overtaking, circular tracks, races and escalators. It is the single largest family in quantitative aptitude: the same core idea generates trains, boats-and-streams and most work-rate questions, so mastering it pays twice. Every Indian campus test uses it — TCS NQT, Infosys, Wipro Elite, Capgemini, Accenture, Cognizant, L&T, bank PO/clerk (IBPS, SBI), SSC, RRB, CAT/XAT — typically 2 to 5 questions, mostly solvable in under a minute with ratio reasoning rather than algebra. Engineering students also meet the same equations as kinematics in physics and as journey-time/cycle-time calculations in logistics, railways and manufacturing.

Explained

Basics, units and proportionality

• Speed is distance covered per unit time: s = d/t. Distance d = s × t, Time t = d/s. These hold exactly only for *uniform* (constant) speed; otherwise s means *average* speed.
• SI unit of speed is m/s; Indian road and rail practice uses km/h; aviation uses knots (1 knot = 1.852 km/h).
• Conversion: 1 km/h = 5/18 m/s and 1 m/s = 18/5 = 3.6 km/h. Multiply by 5/18 to go km/h → m/s; by 18/5 for the reverse.
• The three proportionalities decide almost every shortcut:
  1. Time constant → d ∝ s (faster vehicle covers proportionally more in the same time).
  2. Distance constant → s ∝ 1/t (speeds in ratio a : b means times in ratio b : a).
  3. Speed constant → d ∝ t.
• Always make units consistent *before* calculating: minutes to hours (÷60), metres to kilometres (÷1000). A mixed-unit substitution is the most common source of wrong answers.
• Indian context: under the Motor Vehicles Act, 1988, speed limits are notified by MoRTH and may be lowered by state governments — broadly 120 km/h for cars on expressways, 100 km/h on four-lane divided highways and much lower inside municipal limits; these figures are revised from time to time, so quote them as indicative. Vande Bharat trainsets are designed for 160 km/h but run slower on most sections because of track and signalling limits.

Average speed and a journey in parts

• Average speed = total distance ÷ total time — never the plain average of the speeds, unless the *times* are equal.
• Equal distances at two speeds a and b → average is the harmonic mean 2ab/(a+b). Example: 20 km/h out, 30 km/h back → 2(20)(30)/50 = 24 km/h, not 25.
• Equal distances at three speeds a, b, c → 3abc/(ab + bc + ca). For 20, 30 and 60 km/h this is 108000/3600 = 30 km/h.
• Equal *times* at speeds a and b → (a + b)/2 (arithmetic mean).
• Unequal parts: compute each leg's time separately and divide total distance by total time. There is no shortcut, and attempting one is a trap.
• The average speed always lies between the smallest and largest speed, and is pulled towards the speed held for the longer time, not the longer distance.
• Stoppages: scheduled halts reduce average speed without changing running speed. If a bus runs at S km/h but averages S′ km/h including stops, stoppage time per hour of journey = ((S − S′)/S) × 60 minutes.

Speed ratios, and late-or-early problems

• If speed becomes k times, time for the same distance becomes 1/k times. So at 3/4 of usual speed, time = 4/3 of usual → the person is late by 1/3 of the usual time.
• General result: at (a/b) of usual speed, extra time = ((b − a)/a) × usual time. Reading the fraction correctly is the whole skill.
• Two speeds, two outcomes (late by t₁, early by t₂ — the classic "boy and his school" question):
  • d/s₁ − d/s₂ = (t₁ + t₂), with all times in the same unit → d = (s₁s₂/(s₂ − s₁)) × (t₁ + t₂).
  • Usual/scheduled time then follows by subtracting the lateness from d/s₁.
• If the same distance is covered in times t₁ and t₂ at speeds s₁ and s₂, then s₁t₁ = s₂t₂ — a one-line equation that replaces most algebra here.

Relative speed: meeting, overtaking, gaps

• Relative speed is the speed of one body as seen from the other.
  • Opposite directions (approaching or crossing): relative speed = s₁ + s₂.
  • Same direction (chasing or overtaking): relative speed = |s₁ − s₂|.
• Time to close a gap = gap ÷ relative speed. Everything else is bookkeeping of the gap.
• Two bodies start simultaneously from points d apart and move towards each other: they meet after t = d/(s₁ + s₂), and the meeting point is s₁t from the first point — so the distances covered are in the ratio of speeds.
• Head start in time: if the slower body leaves Δt earlier, its head-start distance is s_slow × Δt; the faster one needs that distance ÷ (s_fast − s_slow) to catch up.
• The √ rule (very frequently asked): two bodies start at the same instant from A and B towards each other, meet, and then take t₁ and t₂ more to reach B and A respectively. Then s₁/s₂ = √(t₂/t₁), and the time to the meeting point is T = √(t₁t₂).
• Repeated journeys between two ends: two bodies starting from opposite ends of a straight path of length d meet for the n-th time after covering a combined distance of (2n − 1)d, so t = (2n − 1)d/(s₁ + s₂).

Circular tracks and races

• Circumference C, speeds a and b from the same starting point:
  • First meeting (same direction) after C/(a − b); first meeting (opposite directions) after C/(a + b).
  • Both back at the start together after LCM of C/a and C/b (take the LCM of the fractions).
  • Writing a : b in lowest terms as p : q, the number of distinct meeting points on the track is p + q (opposite directions) and |p − q| (same direction).
• Races — standard language, worth memorising:
  • "A beats B by 20 m in a 200 m race" → when A finishes, B has run 180 m → speed ratio A : B = 200 : 180 = 10 : 9 (times inverse, 9 : 10).
  • "A beats B by 5 seconds" → B's time − A's time = 5 s.
  • "A gives B a start of 20 m" → B runs only (L − 20) m while A runs L m; "a start of 5 s" → B starts 5 s earlier.
  • Dead heat: both finish together, so the handicap exactly cancels the speed difference.
• In a three-way race, combine ratios: if A : B = 10 : 9 and B : C = 10 : 9, then A : C = 100 : 81, i.e. in a 100 m race A beats C by 19 m.

Escalators, walkways and non-uniform motion

• Moving escalator or walkway (and the same logic as a river current): effective speed = own speed + escalator speed when walking with it, and own speed − escalator speed against it. Count steps or length in the same unit as the speeds.
• If a person takes m steps walking up a moving escalator and n steps walking down it at the same walking rate, the total number of visible steps = 2mn/(m + n) when the up and down walking speeds are equal (and more generally comes from equating the escalator's contribution in both trips).
• Uniformly accelerated motion (physics/GATE overlap, occasionally in aptitude): for initial speed u, final speed v, acceleration a, time t and distance S —
  • v = u + at, S = ut + ½at², v² = u² + 2aS, and average speed = (u + v)/2 only for *uniform* acceleration.
• Limitations of the uniform-speed model: real journeys include acceleration, braking, traffic signals, gradients and halts, so computed times are lower bounds; railway and bus timetables therefore add recovery/slack time. In exams, assume uniform speed unless acceleration is explicitly given.

Key points

• Core relation - speed = distance/time; rearrange, but keep units consistent throughout.
• Conversion - km/h → m/s multiply by 5/18; m/s → km/h multiply by 18/5 (3.6).
• Inverse variation - for a fixed distance, speed ratio a : b gives time ratio b : a.
• Average speed - total distance ÷ total time, never the mean of speeds unless times are equal.
• Harmonic mean - equal distances at a and b average to 2ab/(a+b), always less than (a+b)/2.
• Fraction of speed - at 3/4 speed the time is 4/3 of usual, i.e. 1/3 of usual time late.
• Late-early formula - d = s₁s₂(t₁+t₂)/(s₂−s₁), with t₁+t₂ the total time gap in hours.
• Relative speed - add for opposite directions, subtract for the same direction.
• Catch-up time - head-start distance ÷ difference of speeds.
• Meeting ratio - two bodies starting together towards each other cover distances in the ratio of their speeds.
• √ rule - s₁/s₂ = √(t₂/t₁) and meeting time T = √(t₁t₂) after a head-on start.
• n-th meeting - combined distance (2n−1)d on a straight path between two ends.
• Circular track - first meeting at C/(a±b); both at start again at LCM of lap times.
• Meeting points - p+q (opposite) and |p−q| (same direction) for reduced speed ratio p : q.
• Races - beaten by x metres in an L-metre race → speed ratio L : (L−x).
• Stoppage - ((S − S′)/S) × 60 minutes of halt per hour of travel.
• Escalator/current - add or subtract the medium's speed to get the effective speed.
• Dimension check - a speed answer must come out in distance/time; if it doesn't, a unit was dropped.

Formulas and examples

Example 1 — average speed both ways. A car goes from Kochi to Thrissur at 60 km/h and returns at 40 km/h.

• Equal distances → average = 2(60)(40)/(60+40) = 4800/100 = 48 km/h.
• Check with 120 km each way: 120/60 = 2 h, 120/40 = 3 h; total 240 km in 5 h = 48 km/h ✓ (not 50).

Example 2 — late and early. A student walking at 5 km/h reaches college 6 minutes late; at 6 km/h he is 2 minutes early. Find the distance and the scheduled time.

• Total time gap = 6 + 2 = 8 min = 8/60 = 2/15 h.
• d/5 − d/6 = 2/15 → d(6−5)/30 = 2/15 → d/30 = 2/15 → d = 4 km.
• At 5 km/h: 4/5 h = 48 min, which is 6 min late → scheduled time = 42 min. Check at 6 km/h: 40 min = 2 min early ✓.

Example 3 — overtaking with a head start. A truck leaves a depot at 09:00 at 40 km/h. A van leaves the same depot at 10:30 at 60 km/h on the same road. When and where does the van overtake the truck?

• Head start = 1.5 h × 40 = 60 km. Relative speed = 60 − 40 = 20 km/h.
• Catch-up time = 60/20 = 3 h after 10:30 → 13:30.
• Distance from depot = 60 × 3 = 180 km; truck's check: 4.5 h × 40 = 180 km ✓.

Example 4 — the √ rule. Two buses start at the same moment from Pune and Mumbai (150 km apart) towards each other. After meeting, the Pune bus needs 1 h more to reach Mumbai and the Mumbai bus 2.25 h more to reach Pune. Find both speeds.

• s₁/s₂ = √(2.25/1) = 1.5 → ratio 3 : 2.
• Meeting time T = √(1 × 2.25) = 1.5 h.
• Pune bus: s₁(1.5 + 1) = 150 → s₁ = 60 km/h; Mumbai bus: s₂(1.5 + 2.25) = 150 → s₂ = 40 km/h.
• Check: meeting point 60 × 1.5 = 90 km and 40 × 1.5 = 60 km, total 150 km ✓.

Example 5 — circular track. Two runners start together from the same point on a 400 m track at 5 m/s and 3 m/s.

• Opposite directions: first meeting at 400/(5+3) = 50 s; same direction: 400/(5−3) = 200 s.
• Lap times: 80 s and 400/3 s → both at the start together at LCM(240/3, 400/3) = LCM(240,400)/3 = 1200/3 = 400 s.
• Ratio 5 : 3 → 8 distinct meeting points moving oppositely, 2 moving in the same direction.

Example 6 — race. In a 200 m race A beats B by 20 m, or equivalently by 5 seconds. Find both speeds.

• B covers 20 m in 5 s → s_B = 4 m/s (14.4 km/h) → B's time for 200 m = 50 s.
• A's time = 50 − 5 = 45 s → s_A = 200/45 = 4.44 m/s (16 km/h); ratio 200 : 180 = 10 : 9 ✓.

Where it is used

• Logistics and e-commerce: Delhivery, Blue Dart, Flipkart and Amazon India plan route times, delivery SLAs and vehicle turnaround from average speed and halt time; last-mile planners quantify trips per rider per shift.
• Railways and metro operations: Indian Railways computes sectional running time, headway and crossing time from relative speed; DMRC and other metros set dwell + run times per station pair for timetabling.
• Road transport and fleet telematics: state transport undertakings (KSRTC, MSRTC) schedule buses using running speed versus commercial (with-stops) speed; GPS-telematics dashboards report exactly these two numbers.
• Aptitude-gated hiring: the first screening round at TCS, Infosys, Wipro, Accenture, Capgemini, Cognizant and in IBPS/SBI, SSC and RRB exams; also CAT/XAT quant.
• Industrial engineering and manufacturing: conveyor speed versus line length gives transit and cycle time; AGV travel time inside a plant is the same calculation.
• Civil and traffic engineering: travel-time and journey-speed surveys, signal timing and IRC-based highway capacity studies use journey speed = distance ÷ total time including delays.

Interview questions

• Q: Why is the average of two speeds usually wrong for a there-and-back trip? A: More time is spent at the slower speed, so the correct figure is the harmonic mean 2ab/(a+b), which is always less than (a+b)/2.
• Q: Convert 72 km/h to m/s and 25 m/s to km/h. A: 72 × 5/18 = 20 m/s; 25 × 18/5 = 90 km/h.
• Q: If a man walks at 3/4 of his usual speed, how late is he? A: His time becomes 4/3 of usual, so he is late by 1/3 of his usual time.
• Q: Two people start from the same point and the same time in the same direction at 6 and 4 km/h. When is the gap 5 km? A: Relative speed 2 km/h, so after 5/2 = 2.5 hours.
• Q: State the relative-speed rule for same and opposite directions. A: Subtract the speeds for the same direction, add them for opposite directions; time = gap ÷ relative speed.
• Q: Two cars start simultaneously towards each other and after meeting take 4 h and 9 h to finish. What is the speed ratio? A: √(9/4) = 3 : 2, and they met after √(4×9) = 6 hours.
• Q: On a 500 m circular track, runners at 7 and 5 m/s start together — when do they first meet going the same way? A: 500/(7−5) = 250 seconds.
• Q: A beats B by 10 m in a 100 m race; by how much does A beat B in a 200 m race at the same speeds? A: The speed ratio is 10 : 9, so in 200 m A beats B by 20 m.
• Q: A bus runs at 54 km/h but averages 45 km/h with stops. How long does it stop per hour? A: (54 − 45)/54 × 60 = 10 minutes per hour.
• Q: Why do two bodies starting towards each other cover distances in the ratio of their speeds? A: They travel for the same time until they meet, and for equal time distance is proportional to speed.
• Q: If a train's speed is increased by 25%, by what percentage does journey time fall? A: Speed × 5/4 → time × 4/5, a 20% reduction.
• Q: When does speed = distance/time fail? A: Whenever the speed is not constant — then it gives only the average speed, and instantaneous speed needs the derivative ds/dt or the kinematic equations.

Common mistakes

• Averaging the two speeds instead of using total distance ÷ total time (the harmonic-mean trap) — the most penalised error in the topic.
• Mixing units: substituting minutes with km/h, or metres with km/h, without converting (5/18 and 18/5 get interchanged under time pressure).
• Adding speeds when the bodies move in the same direction, or subtracting when they approach each other.
• In late-or-early problems, using the difference of the two delays instead of their sum when one is late and the other early.
• Misreading the fraction of speed: "4/5 of usual speed" makes the time 5/4 of usual, so the delay is 1/4 — not 1/5 — of the usual time.
• In races, treating "beats by 20 m" as a time difference, or forgetting that the loser's distance is L − 20 while the winner covers the full L.
• On circular tracks, using the LCM of lap times for the *first meeting* — the first meeting is C/(a ± b); the LCM gives the first meeting at the starting point.
• Rounding mid-solution (for example 200/45 to 4.4) and then carrying that rounded value into the next step.

Related topics

Trains, boats and streams · Time and work · Ratio, proportion and partnership · Averages, mixtures and alligation · Percentages · Algebra: linear and quadratic equations