A man is 3/8th of the way across a railroad bridge when he hears a train coming. He knows that this passenger train cruises along at 60 MPH in this area. It turns out that he can run to either end of the bridge and jump off just ahead of the train. Assuming he runs a a constant speed, how fast does he have to run?
To determine how fast the man needs to run to jump off the bridge just ahead of the train, we need to analyze the situation.
Let's consider the distance between the man and the approaching train when he decides to start running as "x." This distance can be calculated as:
x = (3/8) * distance of the bridge
Since the man runs at a constant speed, his velocity can be represented as:
velocity = distance / time
Therefore, the time it takes for the man to reach either end of the bridge and jump off can be expressed as:
time = (distance of the bridge - x) / velocity
We know that the train travels at a speed of 60 MPH, which is roughly 88 feet per second. Therefore, we can substitute the given values into the equation:
time = (distance of the bridge - (3/8) * distance of the bridge) / 88
Simplifying the equation:
time = (5/8) * distance of the bridge / 88
Since we want the man to jump off just ahead of the train, the time it takes for him to reach the end of the bridge should be equivalent to the time it takes for the train to reach his starting position. This can be written as:
time = x / 60
By equating the two expressions for time:
(x / 60) = (5/8) * distance of the bridge / 88
To find the value of x, we multiply both sides of the equation by 60:
x = (5/8) * distance of the bridge * (1/88) * 60
Simplifying the equation:
x = (5/8) * (60/88) * distance of the bridge
x = (5/8) * (15/22) * distance of the bridge
x = (75/176) * distance of the bridge
Finally, the man needs to run a distance of x just before the train reaches his starting position. Therefore, his speed can be calculated by dividing the distance x by the time it takes to reach the end of the bridge:
speed = x / (distance of the bridge / 60)
Substituting x with its value:
speed = [(75/176) * distance of the bridge] / (distance of the bridge / 60)
Simplifying the equation:
speed = (75/176) * 60
Hence, the man needs to run at a speed of approximately 25.68 MPH (rounded to two decimal places) in order to jump off the bridge just ahead of the train.
To find out how fast the man has to run, we need to consider the relative distance covered by the train and the man.
Since the man is 3/8th of the way across the bridge, we can assume that he has covered 3/8th of the distance of the bridge. This means that the remaining distance that he needs to run to reach safety is 1 - 3/8 = 5/8 of the total bridge distance.
Now, let's analyze the time it takes for the train to cover this remaining distance. We know that the train cruises at a speed of 60 MPH. So, if we denote the remaining distance of the bridge as D, the time it takes for the train to cover this distance is D / 60.
Since the man needs to cover the same remaining distance, which is 5/8 of the total bridge distance, his time can be represented as (5/8)D / R, where R is the speed at which the man runs.
To find the speed of the man, we can set up the following equation:
(5/8)D / R = D / 60
Now, we can solve for R:
5/8 = 1 / 60R
Cross-multiplying, we get:
5R = 8 * 60
R = 8 * 60 / 5
R = 96 MPH
Therefore, the man needs to run at a speed of 96 MPH to reach safety and jump off the bridge just ahead of the train.