Question

Determine the average spacing between cars of a row

proceeding at 60 km/h, if the driver of a car travelling
at 80 km/h in the opposite direction passes by 14 cars
of the row per kilometre. (The average length of the
cars is 5 m.)

Answer 1

Average spacing between cars of a row =
`120\textrm{ }m`

Step-by-step explanation:

Let us assume the average spacing between cars to be
`S`.

Frame of Reference : Frame attached to Cars moving at
`60 (km)/(h)`.

In this frame, the cars in the row are at rest, while the car travelling in opposite direction appears to move at a relative speed of
`60 (km)/(h) +80 (km)/(h)=140(km)/(h)`.

He is covering a distance of
`1\textrm{ }km` in ground frame. His speed in ground frame is
`80 (km)/(h)`. So, time taken to traverse 1 km is

`Time taken = \frac{\textrm{Distance covered in that frame}}{\textrm{Speed in that frame }}`

`T = (1 km)/(80 (km)/(h) )`

`T = (1)/(80) h`

In this T time, he moves in the relative frame at
`140`
`(km)/(h)`. The distance covered in relative frame will be

`\textrm{Distance covered in a frame} = \textrm{Speed in that frame}`×
`\textrm{time}`

`Distance = \textrm{140 } (km)/(h)` ×
`(1)/(80) h`

`Distance = (7)/(4) km`

It passes 14 cars while traversing this distance.

∴
`(7)/(4) km =` 14 ×
`(\textrm{Length of a car + Distance between two cars})`

`(7)/(4) km = 14` ×
`(5 m+S)`

`(1)/(8)\textrm{ }km\textrm{ } =\textrm{ } 5\textrm{ }m\textrm{ }+\textrm{ } S`

`S\textrm{ }=\textrm{ }125\textrm{ }m\textrm{ }-\textrm{ }5\textrm{ }m\textrm{ }=\textrm{ }120\textrm{ }m`.

∴Average Spacing betwwen cars in a row =
`120\textrm{ }m`