The acceleration of a 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) t + (1)/(3) bt^(2)`
B. `v_(0) t+ (1)/(3) bt^(3)`
C. `v_(0)t + (1)/(6) bt^(3)`
D. `v_(0) t + (1)/(2) bt^(2)`

Answer

C

Explanation

`a= bt or (dv)/(dy) = bt`. Integrating, we get
`v= (bt^(2))/(2) + c`, where c is a constant of integration
At `t= 0, v= v_(0)`. Thus `v_(0)=c`
Now, `v= (ds)/(dt) = (bt^(2))/(2) + v_(0) :. ds = ((bt^(2))/(2) + v_(0))dt` Integrating we get `s= (bt^(3))/(6) + v_(0)t`