A subway train is stalled in a station. A second train approaches the station at 68.5 km/h and brakes to a halt in 48.3 s, stopping just 1.45 m short of the stalled train. What was the distance between the two trains at the instant the moving train began to brake?

The distance between the two trains at the instant the moving train began to brake was 460.74 m.

Explanation:

Hi there!

The equations of position and velocity of the moving train are the following:

x = x0 + v0 · t + 1/2 · a · t²

v = v0 + a · t

Where:

x = position of the train at a time t.

x0 = initial position.

v0 = initial velocity.

t = time.

a = acceleration.

v = velocity of the train at a time t.

Let's place the origin of the frame of reference at the point where the train starts slowing down so that x0 = 0.

Using the equation of velocity, we can calculate the acceleration of the train:

v = v0 + a · t

When the train stops at t = 48.3 s its velocity is zero. Then:

0 = v0 + a · t

-v0 / t = a

v0 = 68.5 km/h · (1000 m / 1 km) · (1 h / 3600 s) = 19.0 m/s

t = 48.3 s

a = -19.0 m/s / 48.3 s

a = -0.393 m/s²

Now that we have the acceleration, we can find the position of the train after the 48.3 s it takes the train to stop:

x = x0 + v0 · t + 1/2 · a · t²    (x0 = 0 because the origin is the initial position)

x = 19.0 m/s · 48.3 s - 1/2 · 0.393 m/s² · (48.3 s)²

x = 459.29 m

The train traveled 459.29 m and stopped at a distance of 1.45 m from the other train. So, the distance between the two trains at the instant the moving train began to brake was (459.29 m + 1.45 m) 460.74 m.

where are the vectors ?