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.
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.