Question

the velocity of a particle moving in the positive direction of the x axis varies as V =k root s where k is a positive constant. nature of v-t graph
a)parabola. b)hyperbola. c)straight line
give reason

Answer 1

v = k √s
       v is velocity,  k is a positive constant and s is the displacement.
=>  v = ds/dt = k √s
=> ds/√s = k dt
Integrate on both sides, with C as the integration constant.
   =>  2 √s  = k t + C
Let us say that displacement s is 0, at t = 0.  Then, C = 0.
  2 √s = k t
=>  s = (k²/4) * t²

Differentiate wrt t to get velocity time function.
  v = ds/dt = (k²/2) * t

Since velocity is a linear function of t,  the v-t graph is a straight line.
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Another way:

v = k √s        or  s = v² / k

differentiate wrt t:    ds/dt = 2 v dv/dt * 1/k
as  ds/dt = v we get,
 =>  dv/dt = k / 2
 =>  v = (k/2)  t  + C

So veloccity time graph is a straight line.