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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 :-

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 :-

Grade:12th pass

2 Answers

Pawan Prajapati
askIITians Faculty 60787 Points
2 years ago
Our expert is working on this class X maths answer. We will update the answer very soon.
Yash Patari
askIITians Faculty 2123 Points
2 years ago
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

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