|
Acceleration versus velocity graph of a particle moving in a straight line starting from rest is shown in the figure below.
How will the graph representing the variation of velocity with time look like? |
|
The above graph implies that acceleration is decreasing linearly with increase in velocity. This means - v = ma where m is slope of the line given in the figure. But we also know that a = dv/dt. Replace a with v/m, we get v/m = dv/dt => dt = m dv/v Integrate on both the sides t = m In(v) + C [here In v is log with base e] => v will be exponential with respect to t. |
one can also think about it intuitively. acceleration is decreasing with velocity increase. Means that rate of change of velocity is decreasing with increase in velocity linearly. That means velocity with respect to time will be exponential. Of course the mathematical explanation below is more elegant,