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.