The initial velocity of a particle is u and the acceleration is given by (kt), where k is a positive constant. The distance traveled in time t is :-

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DEAR STUDENT

According to the given problem,
- (i) The initial velocity of a particle is u and the acceleration is given by kt, where k is a positive constant.
- (ii) Let D denotes the distance travelled in time t.
As because the acceleration is not constant but a variable as a function of time, we have to use calculus for the current problem. Hence we get,
- (a =) dv/dt = kt or dv = kt dt
- or indefinite-integral [dv] = indefinite-integral [kt dt]
- or v = (k/2)*t^2 + C …… (1a) [C = constant of integration]
- or v = (k/2)*t^2 + u …… (1b)
- [at t=0, from (i) we get v=u, hence from (1a), C=u]
From (1b) we get,
- (v =) ds/dt = (k/2)*t^2 + u
- or ds = [(k/2)*t^2 + u] dt
- or definite-integral [ds] [LL: s=0 & UL: s=D]
- = definite-integral {[(k/2)*t^2 + u] dt} [LL: t=0 & UL: t=t]
- [LL=lower limit, UL=upper limit]
- or D = ut + (1/6)*kt^3 [Ans]
- THANKS