# If a runner can run 100m in 15 seconds into the wind, but running with the wind it takes 10 seconds. What is the speed of the wind?

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When a runner runs against the wind the speed of the runner is reduced by the speed of the runner and the speed of the runner is increased by the speed of the wind when the runner runs in the same direction as the wind.

Let the speed of the runner be R and that of the wind be W. When running in the same direction, the runner takes 10 seconds to run 100 m.

This gives W + R = 100/10 = 10 m/s ...(1)

When running in the opposite direction the runner takes 15 seconds to travel 100 m.

This gives R - W = 100/15 ...(2)

(1) - (2) gives

2*W = 10 - 100/15

=> W = 5 - 100/30 = 1.67

**The speed of the wind is 1.67 m/s**

Let us assume player runs with a velocity V and wind blows in a velocity W.

At the first instance wind blows against the runner. So a velocity of W will act against the direction of runner.This will decrease the average speed the of the runner to (V-W).

At the second instance velocity of the wind helps the player by blowing at the same direction of the runner. So the average speed of the runner is (V+W).

We know that velocity = distance / time

For first instance;

V-W = 100/15------(1)

For second instance;

V+W = 100/10-----(2)

By (2)-(1)

2W = 100/10-100/15

W = 50/15

= 3.3333m/s

**So the wind is blowing at a speed of 3.333 m/s.**

Assumption

- At both cases the player runs with his maximum speed from the start.
- There is no acceleration or deceleration of the player while running.

Let the speed of the runner be x m/s and the speed of the wind is w m/s The runner talkes 15 sec. to run 100 m into the wind. His relative speed in to the wind = x - w which is equal to 100/15 (speed= distance/time). Runner takes 10 sec. to run 100m with the wind, thus his relative speed with the wind = x + w which is equal 100/10. We have obtaind two equationd :

x - w = 100/15 = 20/3 ------- 1) &

x + w = 100/10 = 10 ------- 2)

x - w = 20 /3 -------i)

x + w = 10 -------ii)

Adding equations 1 and 2 , we get :

2x = 50/3

or, x = 50/6 = 25/3 Here we got the value of x = 25/3

substituting value of x in equation 2 ( x + w = 10 )

x + w = 10

or, 25/3+ w =10

or, w = 10 - (25/3) = ( 30 - 25 )/3 = 5/3

therefore w = 5/3 or 1.666... or 1.67

Hence the speed of the wind (w ) = 1.67 m/s **Answer**