The acceleration of particle is increasing linearly with time `t` as `bt`. The particle starts from the origin with an initial velocity `v_0`. The distance travelled by the particle in time `t` will be.

Options

A. `v_(0)+1/3bt^(2)`
B. `v_(0)+1/3bt^(3)`
C. `v_(0)+1/6bt^(2)`
D. `v_(0)+1/2bt^(2)`

Answer

C

Explanation

`a=(dv)/(dt)=bt`
`int_(v_(0))^(v) dv=int_(0)^(t) bt`
`v-v_(0)=(bt^(2))/2`
`v=v_(0)+(bt^(2))/2`
`(ds)/(dt)=v_(0)+(bt^(2))/2`
`int_(0)^(s)ds=int_(0)^(t)(v_(0)+(bt^(2))/2)dt`
`s=v_(0)t+(bt^(3))/6`