Question

A man is on a 1/4 on a bridge. A train is coming the same direction he is going. The man can run across the bridge in the same direction and make in barely in time. He can also run backwards towards the train and also barely make it. How fast is the train going relative to the man.

Answer 1

The train's speed relative to the man is 3 times the man's running speed.

Let's denote the man's running speed as 'm' and the train's speed as 't'.

Since the man can cross the bridge in the same direction as the train, we can assume that the train's speed is faster than the man's running speed. Let 'x' be the distance the man has to cover to get to the other side of the bridge.

According to the problem, the man can barely make it across the bridge if he runs in the same direction as the train. This means that the time it takes the man to cover the distance 'x' is almost equal to the time it takes the train to cover the same distance.

In terms of speed and time, this can be expressed as:

x/m ≈ (x+x)/t

Simplifying the equation:

x/m ≈ 2x/t

Multiplying both sides by m:

x ≈ 2mx/t

Dividing both sides by x:

1 ≈ 2m/t

Solving for t:

t ≈ 2m

Since the man can also run backward towards the train and barely make it, this means that the relative speed of the train and the man is equal to the sum of their speeds.

Relative speed = m + t ≈ m + 2m = 3m

Therefore, the train's speed relative to the man is 3 times the man's running speed.

Answer 2

The train is twice fast going relative to the man.

Explanation:

Consider the provided information.

Let the distance between the train and beginning of bridge is x and length of bridge is y.

A man is on a 1/4 on a bridge. Thus, the 1/4 of y is y/4.

The train is going x distance in time man runs y/4 distance.

Also if the train is going x + y in time man runs the distance 3y/4.

For better understanding refer the figure 1:

So, if train goes x+y-x distance in time man covers the distance 3y/4 - y/4

Now solve 3y/4 - y/4 = 2y/4

The train covers y distance in the time man runs 2y/4 = y/2

That means train covers 2 times of the distance cover by the man or the train goes twice as fast as man.

Hence, train is twice fast going relative to the man.