A particle is moving under constant acceleration `a=alphat+betat^(2)`, where alpha beta constants. If the position and velocity of the partical at start, i.e. `t=0` are `x_(0)` and `v_(0)`, find the displacement and velocity as a function of time t.

Explanation

`a=alphat+betat^(2)`
`(dv)/(dt)=alphat+betat^(2)`
`int_(v_(0))^(v) dv =int_(0)^(t)(alphat+betat^(2))dt`
`|v|_(v_(0))^(v)=|alphat_(2)/2+(betat^(3))/3|_(0)^(t)`
`v-v_(0)=((alphat^(2))/2+(betat^(3))/3)-(0)`
`v=v_(0)+(alphat^(2))/2+(betat^(3))/3`
`(dx)/(dt)=v_(0)+(alphat^(2))/2+(betat^(3))/3`
`int_(x_(0))^(x) dx=int_(0)^(t)(v_(0)+(alphat^(2))/2+(betat^(3))/3)dt`
`|x|_(x_(0))^(x)=|v_(0)t+(alphat^(3))/6+(betat^(4))/12|_(0)^(t)`
`x-x_(0)=v_(0)t+(alphat^(3))/6+(betat^(4))/12`
`x=x_(0)+v_(0)t+(alphat^(3))/6+(betat^(4))/12`