# Concept: Time, Speed and Distance Quick Revision

BASIC CONCEPTS

SPEED

Speed = $\frac{Distance}{Time}$

1 km/hr = $\frac{5}{18}$ m/s

1 m/s = $\frac{18}{5}$ km/hr

PROPORTIONALITY

We have, Distance = Speed × Time

Using the concept of proportionality that we learnt in Variation chapter, we can get some important results.

SOME IMPORTANT RESULTS
• When speed is constant, distance travelled is directly proportional to the time travelled.

i.e., D ∝ T

$\frac{{D}_{1}}{{D}_{2}}$ = $\frac{{T}_{1}}{{T}_{2}}$

• When time travelled is constant, distance travelled is directly proportional to the speed.

i.e., D ∝ S

$\frac{{D}_{1}}{{D}_{2}}$ = $\frac{{S}_{1}}{{S}_{2}}$

• When distance travelled is constant, speed travelled is inversely proportional to the time travelled.

i.e., S ∝ 1/T

$\frac{{S}_{1}}{{S}_{2}}$ = $\frac{{T}_{2}}{{T}_{1}}$

So, if S1, S2, S3 and T1, T2, T3 are the speeds and times for a given body to cover a given distance then it can be inferred that: S1 : S2 : S3 = $\frac{1}{{T}_{1}}$ : $\frac{1}{{T}_{2}}$ : $\frac{1}{{T}_{3}}$

AVERAGE SPEED

Average speed = $\frac{\mathrm{Total Distance Travelled}}{Total Time taken}$

Note: Average speed is not necessarily equal to average of speeds.

Average speed when time travelled is same

When a person travels at 2 or more than two different speeds for same amount of time, average speed is simple average of individual speeds.

For e.g. if Galib travels at speeds of a, b and c kmph for same amount of time, his average speed = $\frac{a + b + c}{3}$.

This is the only case where average speed = average of speeds.

Average speed when distance travelled is same

When a person travels at 2 or more than two different speeds for same amount of distance, average speed is same as harmonic mean of individual speeds.

For e.g. if Galib travels at speeds of a, b and c kmph for same amount of distance, his average speed = $\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$

Note: When a person travels at two different speeds i.e., x and y, for same amount of distance, the average speed is also given by the formula $\frac{2x}{x+y}$

RELATIVE SPEED

Relative speed when two people are moving in

• Opposite directions = sum of their speeds.
• Same direction = difference of their speeds.

TWO PEOPLE MOVING BETWEEN TWO POINTS

IMPORTANT RESULTS
• Two people P and Q start from points A and B respectively towards each other. They keep running back and forth between these two points.

If the ratio of speeds of faster and slower person is less than 2

Time taken to meet for nth time = (2n - 1)$\frac{D}{\left(a + b\right)}$

where, D is the distance between the two end points and a and b are their speeds.

If f/s > 2, then we need to calculate meeting points manually using proportionality of speed and distance.

TIME TO REACH OTHER END AFTER MEETING

If two men A and B start from two different points X and Y respectively towards other end and after meeting each other A takes ta mins to reach the other end while B takes tb mins to reach the other end.

Let the time taken to meet from start is t mins.
Let speeds of A and B be 'a' and 'b' meters/min.

∴ Time to meet(t) = $\sqrt{{t}_{a}×{t}_{b}}$  and
Ratio of Speeds $\frac{a}{b}=\sqrt{\frac{{t}_{b}}{{t}_{a}}}$

TRAINS

These questions deal with moving objects (especially trains) with certain length.

Time for two objects to cross each other = $\frac{Distance to be travelled}{Relative Speed}$

SOME IMPORTANT RESULTS
• The distance traveled by when two trains cross each other is equal to sum of the lengths of the two trains.
• The distance traveled when a train crosses a pole or person is equal to the length of the train.
• The distance traveled when a train crosses a platform is equal to the sum of the length of the train and length of the platform.
• The distance traveled when a train crosses a person standing on the platform is only its own length
• The distance traveled when a train crosses the driver of another train is equal to only its own length

BOATS AND STREAMS

If the speed of the boat in still water known to be “b” and that of the flow of water is known to be “r” then

Upstream speed (SU) = b - r and Downstream speed (SD) = b + r

b = (SD + SU)/2 and r = b = (SD - SU)/2

RACES

LINEAR RACE

Questions related to Linear Race can be solved using the proportionality approach. Look for the distance travelled by two people in same amount of time. Once we know their distances travelled, then we can say

$\frac{Speed of A}{Speed of B}$ = $\frac{Distance travelled by A}{Distance travelled by B}$

IMPORTANT RESULTS
• A beats B by x meters implies that in a race of L meters, when A reaches the finishing line, B is still x meters behind him.
• A beats B by t seconds implies that B reaches the finish line t seconds after A reached the finish line.
• A beats B by x meters of t seconds implies that B covers x meters in t seconds.
• A gives B a start of x meters implies that B is starting the race x meters ahead of the starting point while A starts from the starting point.
• A gives B a start of t seconds implies that A will start running t seconds after B starts.

CIRCULAR RACE

IMPORTANT RESULTS
Two people are running around a circular track of length 'L' at speeds of 'a' and 'b' respectively. Time taken to meet for the first time
• when running in opposite direction
• anywhere on the circle = $\frac{L}{\left(a + b\right)}$
• at the starting point = LCM$\left[\frac{L}{a},\frac{L}{b}\right]$
• when running in same direction
• anywhere on the circle = $\frac{L}{\left(a - b\right)}$
• at the starting point = LCM$\left[\frac{L}{a},\frac{L}{b}\right]$
Three people are running around a circular track of length 'L' at speeds of 'a', 'b' and and 'c' respectively. Time taken to meet for the first time
• when all 3 are running in same direction
• anywhere on the circle = LCM$\left[\frac{L}{\mathrm{|a - b|}},\frac{L}{\mathrm{|a - c|}}\right]$
• at the starting point = LCM$\left[\frac{L}{a},\frac{L}{b},\frac{L}{c}\right]$
• when two (A and B) are running in a direction and third (C) is running in opposite
• anywhere on the circle = LCM$\left[\frac{L}{\mathrm{\left(a + c\right)}},\frac{L}{\mathrm{\left(b + c\right)}}\right]$
• at the starting point = LCM$\left[\frac{L}{a},\frac{L}{b},\frac{L}{c}\right]$
Two people are running around a circular track of length such that simplified ratio of speeds is a : b, the the number of distinct meeting points will be
• when running in opposite direction = a + b
• when running in same direction = |a - b|

ESCALATORS

The concept of escalators is quite similar to Boats and Streams. While water can flow only from a higher altitude to a lower altitude, an escalator can be a moving up escalator or a moving down escalator.

In such questions a man is generally moving on an escalator which is going up or down. Since escalators are involved, distance is measure in steps and speed is measured in steps/second

SOME IMPORTANT RESULTS

Man's speed on still escalator is 's' steps/sec and escalator's speed is 'e' steps/second, then:

• If the man is going up on a moving up escalator, his effective speed = (s + e)
• If the man is going down on a moving up escalator, his effective speed = (s - e)
• If the man is going up on a moving down escalator, his effective speed = (s - e)
• If the man is going down on a moving down escalator, his effective speed = (s + e)

In case of escalators, we can breakdown the total distance travelled into distance travelled by the escalator and the distance travelled by the person himself/herself.

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