Question

An automobile traveling 95 km/h overtakes a 1.30-km-long train traveling in the same direction on a track

parallel to the road. If the train's speed is 75 km/h, how long does it take the car to pass it, and how far will
the car have traveled in this time? What are the results if the car and train are traveling in opposite
directions?
​

Answer 1

1) 234 s, 6.18 km

The position of the car at time t is described as

`x_c(t) = v_c t`

where

`v_c = 95 km/h` is the velocity of the car

The position of the head of the train instead is given by

`x_t(t) = d + v_t t`

where

d = 1.30 km is the initial distance between the car and the head of the train

`v_t = 75 km/h` is the velocity of the train

The car overtakes the train when

`x_c =x_t`

Substituting and solving for t,

`v_c t = d + v_t t\\t(v_c-v_t)=d\\t=(d)/(v_c-v_t)=(1.30)/(95-75)=0.065 h = 234 s`

And the distance travelled by the car is

`x_c = v_c t = (95 km/h)(0.065 h)=6.18 km`

2) 27.5 s, 0.72 km

In this case, the train is travelling in opposite direction, so we can write

`v_t = -75 km/h`

Again, we can use the same equation as before

`x_c =x_t`

And solving for t, we find

`v_c t = d + v_t t\\t(v_c-v_t)=d\\t=(d)/(v_c-v_t)=(1.30)/(95+75)=0.0076 h = 27.5 s`

And the distance travelled by the car is

`x_c = v_c t = (95 km/h)(0.0076 h)=0.72 km`