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# Relative Speed

Relative speed means comparing the speed of one body to another. Take an example – a body is stationary concerning us. However, the same body may be moving concerning someone who is on the Moon. Therefore, this is how a relative movement is done. In the same way, relative speed also works. For example, two trains going in the same direction have a more considerable relative speed than those going together in the same direction. This is because the speeds add up whenever they go in different directions and reduce whenever they go in the same direction.

We can use the concept of relative speed to find the time when the two trains will meet. For example, consider X and Y starting simultaneously, traveling in the opposite direction with a speed of 20 kmph and 30 kmph, and they have to travel a distance of 200 km.

Two trains moving towards each other are X and Y, with 20 kmph and 30 kmph. Now make train X stationary and take the speed of Y about X. As they are traveling in opposite directions, relative Speed of Y = sum of their respective speeds = 20 + 30 = 50 kmph. Now X can be assumed to be stationary, and the total distance has to be traveled by Y, which it will do with a Speed = 50 kmph. So Y will meet X after 200/50 = 4 hrs.

Example 1
A girl rows a boat at a speed of 15 kmph upstream and 20 kmph downstream. Find the speed with which the girl rows the boat in still water and the stream's speed.
Solution

Given that upstream Speed = 15 kmph
Downstream Speed = 20 kmph
Speed of the girl in still water = x = ((a + b))/2= (20+15)/2 = 35/2
Speed of stream =y = ((a – b))/2= (20-15)/2 = 5/2.

Example 2
The speed of the Godavari River is 5 kmph. A stationary body is placed in the river. Find the time taken by the floating body to reach a stone 10 km downstream from the point where it is now?
Solution

Speed of body = Speed of river (as Speed of boy is 0) = 5 kmph
Speed=Distance/Time.
So, Time taken to reach 10 km = 10/5 = 2 hours.

Example 3
Train A, which is 125 m long, is traveling at 108 km/hr. On a parallel track, a 180 m long Train B travels at 72 km/hr in the same direction. In Train B, a passenger is walking towards the rear end of the train at a speed of 9 km/hr. In how many seconds will the train ultimately cross the passenger in Train B?

(a) 14.4
(b) 10
(c) 16.67
(d) 24
Solution
Speed of A = 108 km/hr = 30 m/s
Speed of train B = 72 km/hr = 20 m/s
Here the relative speed between train A and train B should be considered first i.e.= 30 -20 = 10 m/s.
Relative Speed -Train A and man moving in the opposite direction with a Speed of 2.5 m/s=10 +2.5 = 12.5m/s
Time taken by train A to pass the man who is moving in the opposite direction = 125/12.5 = 10 secs since the distance to be covered over here is the length of train A only.

Example 4
How much time a 100m long train traveling at a speed of 8 m/s will take to overtake another train which is 80m, and is traveling in the same direction at a speed of 4 m/s?
Solution

Suppose the 80 m long train is stationary and take the speed of the 100m long train relative to the former. So relative Speed of 100m long train = 8 – 4 = 4 m/s. To overtake the 80m long train, it has to travel a total Distance = sum of its length and that of the other train = 100 m + 80 m = 180 m. So to travel that distance with a Speed of 4 m/s, it will take 180/4= 45 secs.

Example 5
Two vehicles are traveling from the exact location at 6 km/hr and 4 km/hr, respectively. Calculate the distance between the vehicles after 10 minutes, given that both vehicles are traveling in the same direction.
Solution

The relative speed of the vehicles when they move in the same direction
= (6 – 4) km/hr
= 2 km/hr
Total time taken = 10 minutes
Therefore distance travelled = speed × time
= (2 × 10/60) km
= 1/3 km
= 1/3 × 1000 m
= 333.3 m
Therefore, the distance between the vehicles after 10 minutes is 333.3 m, given that both vehicles are traveling in the same direction.

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