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Time & Speed

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CAT 2025 Lesson : Time & Speed - Concepts & Cheatsheet

Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

12. Cheatsheet

1) When a distance

\space

\space

is covered at a speed of

\space

\space

at time

\space

T, then

\space

D \space = \space S \times T, \space S \space = \space \dfrac{D}{T}

and

T \space = \space \dfrac{D}{S}

\space \space \space \space \space \space

1.1) Distance and Speed have a direct relationship, i.e.

D \propto S

\space \space \space \space \space \space

1.2) Distance and Time have a direct relationship, i.e.

D \propto T

\space \space \space \space \space \space

1.3) Speed and Time, however, have an inverse relationship, i.e.

S \propto \dfrac{1}{T}

\space \space \space \space \space \space \space \space \space \space

1.3.1) If ratio of speed is

S_1 : S_2 : S_3

, then ratio of time taken is

\dfrac{1}{S_1} : \dfrac{1}{S_2} : \dfrac{1}{S_3}

1 \space km/hr = \dfrac{5}{18} \space m/s

and

1 \space m/s = \dfrac{18}{5} \space km/hr

3) Average Speed =

\dfrac{Total \space Distance \space Covered}{Total \space Time \space Taken}

\space \space\space \space \space \space

3.1) When Time Taken are equal, then the average speed is the Arithmetic Mean of the Speeds.

\space \space \space \space \space \space

3.2) When Distances are equal, then the average speed is the Harmonic Mean of the Speeds.

4) When Distance is constant,

\space \space \space \space \space \space\bullet

if one of time or speed increases by

\dfrac{a}{b}

of itself, then the other decreases by

\space \space \space \space \space \space \space \dfrac{a}{b + a}

of itself

\space \space \space \space \space \space\bullet

if one of time or speed decreases by

\dfrac{a}{b}

of itself, then the other increases by

\space \space \space \space \space \space \space \dfrac{a}{b - a}

of itself

5) Where

S_1

and

S_2

are two speeds with

S_1 \space > \space S_2

\space \space \space \space \space \space\bullet

in opposite directions, Time taken

=\space \dfrac{Distance}{Relative \space Speed} \space = \space \dfrac{D}{S_1 + S_2}

\space \space \space \space \space \space\bullet

in same direction, Time taken

=\space \dfrac{Distance}{Relative \space Speed} \space = \space \dfrac{D}{S_1 - S_2}

6) If a train of length l takes time t to pass an

\space \space \space \space \space \space\bullet

object of negligible length at a speed s, then

l = s \times t

\space \space \space \space \space \space\bullet

object of length

l_2

at a speed

s

, then

l_1 + l_2 = s \times t

7) If a train of length

l

moving at speed

s_1

1 takes time

t

to pass an

\space \space \space \space \space \space\bullet

object of negligible length moving at speed

s_2

, moving in opposite direction,

\space \space \space \space \space \space \space l = (s_1 + s_2) \times t

\space \space \space \space \space \space\bullet

object of negligible length moving at speed

s_2

, moving in same direction, then

\space \space \space \space \space \space \space l = (s_1 - s_2) \times t

8) If two trains of length

l_1

and

l_2

moving at speeds of

s_1

and

s_2

respectively, take time

t

to cross each other

\space \space \space \space \space \space\bullet

in opposite directions, then

l_1 + l_2 = (s_1 + s_2) \times t

\space \space \space \space \space \space\bullet

in same direction with

s_1 > s_2

, then

l_1 + l_2 = (s_1 - s_2) \times t

9) Two people walking to and fro from two ends of a corridor of length

l

, when meeting for the

n^{th}

time, together cover a distance of

(2n - 1)l

10) Two people running around circular track at speeds of

s_1

and

s_2(s_1 > s_2)

meet for

n^{th}

time,

\space \space \space \space \space \space \bullet

when running in opposite directions, time taken

= \dfrac{nl}{s_1 + s_2}

\space \space \space \space \space \space \bullet

when running in same directions, time taken

= \dfrac{nl}{s_1 - s_2}

11) Where

s

is the speed of the boat in still water and

a

is the speed of the stream, the relative speed while

\space \space \space \space \space \space \bullet

travelling downstream

= s + a

\space \space \space \space \space \space \bullet

travelling upstream

= s - a

12) Where

x

and

y

are downstream and upstream speeds, boat speed

= \dfrac{x + y}{2}

and stream speed

= \dfrac{x - y}{2}

13)

\space \dfrac{Downstream \space Time}{Upstream \space Time} \space = \space \dfrac{Upstream \space Speed}{Downstream \space Speed}

14) The angles covered by the minute and hour hand for 1 hour and 1 minute are provided below.

$\space \space \space \space \space \space \space \space \space \space \space$	Minute Hand	Hour Hand
1 Hour	$360^0$	$30^0$
1 Minute	$6^0$	$0.5^0$

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