(a) The position of a particle moving along the x-axis depends on the time according to equation. `x=(3t^(2)-t^(3))` m. At what time does the particle reaches its maximum x-position?
(b) What is the displacement during the first 4s.

Answer

3

Explanation

`(dv)/(dt)=v^(2)/RrArr underset(v_(0))overset(v)(int)(dv)/v^(2)=1/Runderset(0)overset(t)(int)dt rArr (-1/v)_(v_(0))^(v)=t/RrArr v=v_(0)/(1-v_(0)/Rt)rArr (ds)/(dt)=v_(0)/(1-v_(0)/Rt)rArr underset(0)overset(2piR)(int)ds=underset(0)overset(t)(int)v_(0)/((1-v_(0)/Rt))dt`
`rArr 2pi R=-R[ln(1-v_(0)/Rt)]_(0)^(t)rArr 2pi=-ln(1-v_(0)/Rt)rArr 1-v_(0)/Rt=e^(-2pi)rArr t=R/v_(0)(1-e^(-2pi))`
`rArr alpha=1, beta=2 rArr (alpha+beta)=(1+2)=3`