Velocity & acceleration

Average velocity:

The ratio of change in displacement to the change in time.

\[v_{avg} = \frac{x_2-x_1}{t_2-t_1} = \frac{\Delta x}{\Delta t}\]

The velocity of an object at a particular instant.

At an instant, \(\Delta t = 0\). Hence, take the limit of average velocity as \(\Delta t \rightarrow 0\).
This is also known as derivative of \(x\) with respect to \(t\).

\[v=\lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dy}\]

In uniform motion, average velocity \(=\) instantaneous velocity

Total change in velocity by the total change in time.

\[\overline{a} = \frac{v_2-v_1}{t_2-t_1} = \frac{\Delta v}{\Delta t}\]

The acceleration of the particle at a particular instant.

\[a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}\]