A bus is beginning to move with an acceleration of 1 m/s^2. A boy who is 48 m behind the bus starts running  at 10 m/s. After what time will the boy be able to catch the bus? Let us say when boy catch the bus the bus has run xm from its initial position. So at that time boy runs (48+x)m. Let us say the time taken to catch the bus is t.

For bus

`S = ut+1/2xxaxxt^2`

`x = 1/2xx1xxt^2 = t^2/2`

`x = t^2/2 ----(1)`

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Let us say when boy catch the bus the bus has run xm from its initial position. So at that time boy runs (48+x)m. Let us say the time taken to catch the bus is t.

For bus

`S = ut+1/2xxaxxt^2`

`x = 1/2xx1xxt^2 = t^2/2`

`x = t^2/2 ----(1)`

For boy

`S = ut+1/2xxaxxt^2`

`48+x = 1/2xx10xxt^2`

`48+x = 5t^2 ---(2)`

(2)-(1)

`48 = 5t^2-t^2/2`

`96 = 9t^2`

`t^2 = sqrt(96/9)`

`t = 3.265s`

So the boy will catch the bus after 3.265 seconds.

Assumptions

• Both boy and bus start the motion from rest.
• Both bus and boy start the motion simultaneously