Understanding the Train and Car Problem

This problem asks us to find the time duration required for a train, moving at a certain speed, to completely overtake a car moving on a parallel track. We are given the speeds of both the train and the car, the length of the train, and crucially, that they are traveling in opposite directions.

Key Information Provided

• Speed of the train ($S_t$): 86 km/h
• Speed of the car ($S_c$): 40 km/h
• Length of the train ($L_t$): 490 m
• Direction of motion: Opposite directions
• Goal: Calculate the time (in seconds) for the train to completely cross the car.

Calculating Relative Speed

When two objects move towards each other (in opposite directions), their speeds add up from the perspective of one object relative to the other. This combined speed is known as the relative speed.

The formula for relative speed in this scenario is:

$S_{relative} = S_t + S_c$

Substituting the given values:

$S_{relative} = 86 \text{ km/h} + 40 \text{ km/h} = 126 \text{ km/h}$

Converting Speed to Meters per Second (m/s)

Since the length of the train is provided in meters (m) and we need the final answer in seconds (s), it's essential to convert the relative speed from kilometers per hour (km/h) to meters per second (m/s).

The conversion factor is $1 \text{ km/h} = \frac{5}{18} \text{ m/s}$.

Applying this conversion to the relative speed:

$S_{relative} (\text{in m/s}) = 126 \text{ km/h} \times \frac{5}{18}$

To simplify the calculation:

$S_{relative} = \frac{126}{18} \times 5$

We know that $126 \div 18 = 7$. So,

$S_{relative} = 7 \times 5 = 35 \text{ m/s}$

Therefore, the relative speed at which the train and car are approaching each other is 35 m/s.

Determining the Distance for Complete Crossing

For the train to cross the car completely, the train must travel a distance equal to its own length relative to the car. Since the length of the car is not provided, we treat it as a point object for the purpose of calculating the crossing time. The distance that needs to be covered is the length of the train itself.

Distance ($D$) = Length of the train ($L_t$) = 490 m

Calculating the Time Taken

The time required to cover a certain distance at a constant speed is given by the formula:

Time ($T$) = Distance ($D$) / Speed ($S$)

Using the distance (length of the train) and the relative speed we calculated:

$T = \frac{D}{S_{relative}}$

Substituting the values:

$T = \frac{490 \text{ m}}{35 \text{ m/s}}$

Performing the division:

$T = 14 \text{ s}$

The train will take 14 seconds to cross the car completely.