## 2440.93 – Train Speed

The train roars through the station. A waiting passenger times its transit past the point where she stands. It takes $5$ seconds. To pass the whole station requires $19$ seconds from the time the engine enters the station until the time the last car exits it. (The passenger didn't know this, but we do.) The platform is $280$ meters long.

How long is the train and what is its speed?

## Solution

In the figure below, P is the location of the passenger; S and E are the start and end (respectively) of the station platform.

Now, it took $19$ seconds from the moment when the front of the train reached S to the moment when the rear of the train passed E. We also know that it took $5$ seconds from the moment when the front of the train passed P to the moment when the rear of the train passed P. Naturally that $5$ seconds applies to any other point on the platform, say S. So it took $14 = 19-5$ seconds for the back of the train to travel the $280$ meters of the platform. Ha! The speed of the train is $\frac{280}{14} = 20 \text{ meters/second}$.

If it takes $5$ seconds for the whole train to pass a point while going $20$ m/s, then the train is $100$ m long.

If you convert this speed to kilometers or miles per hour, you'll see that it's not a particularly fast train. But it still roared.