The average velocity tells us how fast an object has been moving over a given time interval but does not tell us how fast it moves at different instants of time during that interval. For this, we define instantaneous velocity or simply velocity \(v\) at an instant \(t\).
Instantaneous Velocity
The velocity at an instant is defined as the limit of the average velocity as the time interval \(\Delta t\) becomes infinitesimally small. In other words,
\(
v=\lim _{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t} \dots(3.3a)
\)
\(
=\frac{\mathrm{d} x}{\mathrm{~d} t} \dots(3.3b)
\)
which is the It is the rate of change of position with respect to time, at that instant.
Instantaneous speed
Instantaneous speed or simply speed is the magnitude of velocity. So the magnitude of the velocity is
\(
v=\left|\frac{d x}{d t}\right|
\)
which is the instantaneous speed at time \(t\). Instantaneous velocity is also called the “velocity”.
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