The velocity-time relationship which has
been derived graphically earlier, can also be obtained with the help of
calculus.
Acceleration is the rate of change of velocity, and therefore, can be
expressed as
Integrating both sides of the above equation and applying the limits (Which are at t = 0, v(t) = v(0), the initial velocity and after an interval of time = t, the velocity is v(t)), we have,
Position-time relation by method of calculus
Starting with the definition of velocity,
However, v (t) = v (0) + at
dx = v (0) dt + at dt
Integrating the above equation and applying the following limits, i.e., at t =0, x (t) = x (0), after an elapsed time t, the displacement is x (t), we have,
Position-time graphs
From the position-time relationship it is clear that the relation between position and time is a quadratic one. Hence, the position-time graphs for both uniformly accelerated and retarded motions are parabolas, as shown in the figures.
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dx = v (0) dt + at dt
it is clear that the relation between position and time is a quadratic one. Hence, the position-time graphs for both uniformly accelerated and retarded motions are parabolas, as shown in the figures.
