Question

Variable acceleration?

The acceleration a ms-2 of a particle after t seconds is given by a=t. The particle is initially at the origin at rest.
The displacement of the particle from the origin is 4.5m after ____m

Answer 1

Given acceleration `a=t\ ms^-2`
Rewriting it as differential of velocity with respect to time and integrating both sides with respect to time we get

`dotv=t`
`int\ dotv\ dt=int\ t\ dt`
`=>v= t^2/2 +C` .......(1)
where `C` is constant of integration to be found from initial conditions.

It is given that the particle is initially at the origin at rest. Inserting in (1) we get `C=0`. (1) becomes

`=>v= t^2/2`

Rewriting it as differential of displacement with respect to time and integrating both sides with respect to time we get

`int\ dots\ dt= int\ t^2/2\ dt`
`s= t^3/6+C_1` .......(2)
where `C_1` is constant of integration to be found from initial conditions.

Given that initially the particle is at origin. Inserting in (2) we get `C_1=0`. (2) becomes

`s=t^3/6`

Finding the time when displacement of the particle is `4.5\ m` from the origin.

`4.5=t^3/6`
`=>t^3=4.5xx6`
`=>t=root 3(27)`
`=>t=3\ s`
`=>t=3/60=0.05\ min`