Question

A car moving at 95 km/h passes a 1.00-km-long train traveling in the same direction on a track that is parallel to the road. If the speed of the train is 75 km/h, how long does it take the car to pass the train, and how far will the car have travelled in this time? What are the results if the car and train are instead traveling in opposite directions?

Short answer

The obtained values of time and distance for the same direction of the car and train are $t_1=180\mathrm{s}$and $D=4.75\mathrm{km}$, whereas the obtained values of time and distance for the opposite direction of the car and train are $t_2=21.17\mathrm{s}$and $D\text{'}=0.55\mathrm{km}$, respectively.

Step-by-step solution

Step 1. Concept of relative velocity

In this problem, the frame of reference in which the train is steady is considered. First, for calculating the time to pass the train, you will have to find the relative velocity of the car to the train and then use the constant horizontal velocity relation to determine the time required.

Given data:

The velocity of the car is $v_1=95\mathrm{km}/\mathrm{h}$.

The length of the train is $d=1\mathrm{km}$.

The velocity of the train is role="math" localid="1644923596041" $v_2=75\mathrm{km}/\mathrm{h}$.

When the car and train are moving in the same direction, the formula to calculate the time can be written as:

$v_x=\frac{d}{t_1}$

Here, $v_x$is the velocity of the car relative to the train, and $t_1$is the time taken by the car.

On plugging the values in the above relation, you get:

$\begin{array}{c}{v}_1-v_2=\left(\frac{1\mathrm{km}}{t_1}\right)\\ 95\mathrm{km}/\mathrm{h}-75\mathrm{km}/\mathrm{h}=\left(\frac{1\mathrm{km}}{t_1}\right)\\ t_1=\left(0.05\mathrm{h}\times \frac{60\times 60\mathrm{s}}{1\mathrm{h}}\right)\\ t_1=180\mathrm{s}\end{array}$

Thus,$t_1=180\mathrm{s}$ is the time taken by the car to pass the train.

Step 2. Determine the distance traveled by the car

The formula to calculate the distance traveled can be written as:

$D=v_1{t}_1$

On plugging the values in the above relation, you get:

$\begin{array}{c}D=\left(95\mathrm{km}/\mathrm{h}\right)\left(0.05\mathrm{h}\right)\\ D=4.75\mathrm{km}\end{array}$

Thus, $D=4.75\mathrm{km}$is the distance covered by the car to pass the train.

Step 3. Calculate the time taken by the car

When the car and train are moving in opposite directions, the formula to calculate the time can be written as:

$v_x\text{'}=\frac{d}{t_2}$

Here, $v_x\text{'}$is the velocity of the car relative to the train, and $t_2$is the time taken by the car.

On plugging the values in the above relation, you get:

$\begin{array}{c}{v}_1+v_2=\left(\frac{1\mathrm{km}}{t_2}\right)\\ 95\mathrm{km}/\mathrm{h}+75\mathrm{km}/\mathrm{h}=\left(\frac{1\mathrm{km}}{t_2}\right)\\ t_2=\left(\frac{1}{170}\mathrm{h}\times \frac{60\times 60\mathrm{s}}{1\mathrm{h}}\right)\\ t_2=21.17\mathrm{s}\end{array}$

Thus, $t_2=21.17\mathrm{s}$is the time taken by the car to pass the train.

Step 4. Determine the distance traveled by the car

The formula to calculate the distance traveled can be written as:

$D\text{'}=v_1{t}_2$

On plugging the values in the above relation, you get:

$\begin{array}{c}D\text{'}=\left(95\mathrm{km}/\mathrm{h}\right)\left(\frac{1}{170}\mathrm{h}\right)\\ D\text{'}=0.55\mathrm{km}\end{array}$

Thus, $D\text{'}=0.55\mathrm{km}$is the distance covered by the car to pass the train.