A body starts from rest and moves with an acceleration proportionaI to time. (i) Find its velocity n s after starting. (ii) What distance will it traveI in n s?

Explanation

According to the problem `a prop t`
or,`" "` a= kt, where k is the constant of proportionality.
Now, `a = (dv)/(dt) :. Dv = kt dt " ""or, int dv = k int t dt`
`:. " " v = (kt^(2))/(2) +A` [where A = integration constant ] `cdots (1)`
Also, as v = 0 at t =0, we get A = 0
`:.` From equation (1) we get ,
`v = (kt^(2))/(2)`
Again if the displacement of the object is s then `v = (ds)/(dt)`
`:.` From equation (2)
`(ds)/(dt) = (kt^(2))/(2) "or", ds =(1)/(2) kt^(2)dt`
On integration we get ,
`s = (kt^(3))/(6) + B` [ where B = integration constant ]
Also, at t = 0, s = 0 `:.` B = 0
`:. s = (kt^(3))/(6)`
At t = n s, from equations (2) and (3) we get,
(i) `v= (kn^(2))/(2) ` and (ii) `s = (kn^(3))/(6)`.