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CAT 2009 Online Preparation

Time, Speed & Distance # 3

Author: gauravjain26

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Related speed And Proportion Related to Relative Speed

As the name suggests, Relative speed refers to the speed of one object with respect to another object. Another object can be:

Mmoving in the same direction.

Moving in the opposite direction.

Stationary.

In the above two conditions both the objects move; where as in third condition only one object moves.

POSSIBLE CASES

CASE # 1:

When an object is moving and other is stationary.

consider a train and a Milestone pillar.Here train is moving and the pillar is stationary.

Relative speed of train = speed of train

CASE # 2:

When two objects are moving in same direction.

Consider two trains standing parallel to each other and start moving in same direction.

Relative speed of two objects = Difference of their relative speed.

CASE # 3:

When two objects are moving in opposite direction.

Consider two trains standing opposite to each other and starts moving towards each other.

Relative speed of two objects = Sum of their relative speed.

SOME IMPORTANT POINTS

Time taken by a moving object, say A, of 'x' meter length in passing a stationary object of negligible length, say a pillar or a man or a milestone, is same as the time taken by A to cover'x'meters with it's own speed.

Say, Train's length is 10 meters and speed is 10 meters/minute and it crosses a pillar. In that case it will take 1 minutes to train to pass the pillar completely.

Time taken by a moving object, say A, 'x' meters long in passing a stationary object, say B, 'y' meters long is same as the time taken by A in covering the distance (x + y) with it's own speed.

Say, train's length is 10 metes and speed is 10 meters/minute and it crosses a build, say cleaning room, 10 meters long. In that case it will take 2 minutes to train to pass the pillar completely

.

3)

If two objects of length 'x' and 'y' meters move in the opposite directions at 'a' and 'b' meters/minute speed, then the time taken by these two objects to cover each other completely is:

$= \frac{Sum of Length}{Relative speed} = \frac{x + y}{a + b}$

4)

If two objects of length 'x' and 'y' meters move in the same direction at 'a' and 'b' meters/minute speed, then the time taken by these two objects to cover each other completely is:

$= \frac{Sum of Length}{Relative speed}$
$= \frac{x + y}{a - b}$ ; if a > b
$= \frac{x + y}{b - a}$ ; if b > a

5)

If the speed of a boat in still water is 'a' km/hr and the speed of stream is 'b' km/hr then:

Speed of boat downstream (in same direction/ with stream) =

km/hr.

Speed of boat upstream (against direction/ against stream) =

km/hr; if x > y.
(y - x)

km/hr; if y > x.

Speed of boat in still water
= x

km/hr
$= \frac{1}{2} *$ (speed of boat with stream + Speed of boat against stream)km/hr

Speed of stream/river =* $\frac{1}{2} *$ (speed with stream - speed against stream)km/hr

SOME SOLVED EXAMPLES

EXAMPLE # 1:

If A goes to market at 12 km per hour but could manage to return only at 6 km/hr. Find A's average speed.

SOLUTION:
$Average Speed = [\frac{2ab}{a + b}]\frac{km}{hr}$

$Average Speed = \frac{2 * 12 * 6}{12 + 6} = \frac{144}{18} = 8$

EXAMPLE # 2:

Walking at $\frac{3}{4}th$ of the usual speed, Mr. X is 2 hours late. Find his usual travel time.

SOLUTION:
Suppose usual time is 't' hr and usual speed is 'x' km/hr. Then
Distance Traveled = (Speed) * (Time taken)

Distance Traveled = (Speed) * (Time taken)

------equation#1 (by usual speed)
$Distance Traveled = \frac{3}{4}x * (t + 2)$ ------equation#2(by new speed)

COMPARING EQUATION 1 AND 2

$t = \frac{3}{4}(t + 2)$

solving this, we get:

t = 6

usual travel time = 6 hr.

Example # 3:

Mr. X traveled 120 km by vehicle A, 450 km by vehicle B and 60 km by vehicle c. The whole journey took 13 hours and 30 minutes. Speed of vehicle B is 3 times the speed of vehicle C and 1.5 times the speed of vehicle A. Find the speed of vehicle B.

SOLUTION:
Suppose speed of vehicle C is x km/hr; then as per question
Speed of vehicle B is 3x km/hr
and Vehicle A is $\frac{Speed of vehicle B}{1.5} = 2x$ km/hr

Total time taken is 13 hours and 30 minutes = $\frac{27}{2}$ hr

Total time taken = time taken in traveling by vehicle A + time taken in traveling by vehicle B + time taken in traveling by vehicle C

$\frac{120}{2x} + \frac{450}{3x} + \frac{60}{x} = \frac{27}{2}$
$\frac{60}{x} + \frac{150}{x} + \frac{60}{x} = \frac{27}{2}$
$\frac{270}{x} = \frac{27}{2}$
x = 20