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Maths / Polynomials / Problems Based on Speed

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Problems Based on Speed

Solving Statements Involving Speed:

In these types of problems we consider the relative speed of the two objects. When the two objects are moving in the same direction, the relative speed is the difference of the speed of the two objects. When the two objects are moving in the opposite direction , the relative speed is found as the sum of speed of the two objects.

Some Important Facts:

Formula 1. When a person changes his speed to ( x/y) of its speed and reaches early by a min, then usual time taken by the person to reach his destination is $left ( frac{a}{1-frac{y}{x}}right ) min.$

Formula 2. When a person changes his speed to (x/y) of its usual time taken by the person to reach the destination is $left ( frac{a}{frac{y}{x}-1} right )$ min.

Formula 3. When a person covers a certain distance between two places with speed a, he gets late by time $t_1$ and when he covers the same distance with speed b, he reaches his destination early by time $t_2,$ then the distance between two places is $frac{ab(t_1+t_2)}{b-a}$ .

Formula 4. When two persons/ trains / cars / objects start traveling towards each other from two distinct points at the same time and complete their journey between the points after passing each other in x and y time respectively, then the ratio of their speed is $sqrt{y}:sqrt{x}.$

Illustration: A train travels a distance of 300 km at a certain constant speed. If the speed of train is increased by 5 km an hour it takes two hours less . Find the original speed of train.

Solution: Let the speed of train = x km/h

distance= 300km

$Speed = frac{distance}{time}$

$time = frac{distance}{speed}$

$time_{original} = frac{300}{x}$

New Speed= x+5

$time_{new} = frac{300}{x+5}$

According to Question

$time_{old}-time_{new} = 2$

$frac{300}{x}- frac{300}{x+5}=2$

$300left [ frac{1}{x}- frac{1}{x+5} right ]=2$

$left [ frac{x+5-x}{x(x+5)} right ]=frac{2}{300}$

$left [ frac{5}{x^2+5x} right ]=frac{1}{150}$

$750=x^2+5x$

$x^2+5x-750=0$

$x^2+30x-25x-750=0$

$x(x+30)-25(x+30)=0$

$(x+30)(x-25)=0$

$either; x= -30 ;or; x=25$

As speed can not be negative so the speed= 25 km/h

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