Instantaneous Velocity

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Instantaneous velocity is the velocity of an object at a particular instant of time. For average velocity, the change in displacement is taken over a period of time, but for instantaneous velocity, displacement is taken at that particular instant of time. Therefore to find the instantaneous velocity, we find the derivative of the displacement function with respect to time so that we get the change in the motion for a very small interval of time. Instantaneous velocity is also a vector quantity as it has both magnitude and direction.

Example 2: Find the instantaneous velocity of an object moving with a displacement function of x = 3t2 at time t = 4secs.

To calculate the instantaneous velocity, we find the derivative of ‘x’ with respect to time, t.
Instantaneous velocity: v = d(x)/dt

Power rule of the Derivatives: df(tn)/ dt = n * tn-1

Given the displacement function, x = 3t2 ==> v = d(3t2)/dt

Applying the derivative formula we get: v = 3 * d(t2)/dt = 3 * (2 * t2-1) = 6t

So at time, t = 4secs ==>Instantaneous velocity, v = 6 * 4 = 24m/sec

Example 2: Find the instantaneous velocity of an object moving with a displacement function of x = t2 – 2t at time t = 3secs.

Instantaneous velocity: v = d(x)/dt

Power rule of the Derivatives: df(tn)/ dt = n * tn-1

Given the displacement function, x = t2 – 2t ==> v = d(t2)/dt –d(2t)/dt

Applying the derivative formula we get: v = (2 * t2-1) - 2 * t1-1 = 2t - 2

So at time, t = 3secs ==>Instantaneous velocity, v = (2 * 3) - 2 = 4m/sec