1. ## Rectilinear Motion

Can someone explain what average velocity is? I know that it's change in position/change in time = $\displaystyle \frac{\Delta s}{\Delta t}=\frac{f(t+\Delta t)-f(t)}{\Delta t}=v_\text{avg}$, but exactly does this mean. Also, how is this different from instantaneous velocity? How is this different from a body's velocity at an exact instant?

2. Originally Posted by Winding Function
Can someone explain what average velocity is? I know that it's change in position/change in time = $\displaystyle \frac{\Delta s}{\Delta t}=\frac{f(t+\Delta t)-f(t)}{\Delta t}=v_\text{avg}$, but exactly does this mean. Also, how is this different from instantaneous velocity? How is this different from a body's velocity at an exact instant?

The difference between average velocity and instanteous velocity is in your $\displaystyle \Delta t$. For average velocity, $\displaystyle \Delta t$ is some fixed amount, like $\displaystyle \Delta t = 1$ or $\displaystyle \Delta t = 5$ whereas for instanteous velocity, $\displaystyle \Delta t \rightarrow 0$. For example, suppose you are given the postion function.

$\displaystyle s(t) = 16 t^2 \;ft$

Then the average velocity going from 1 to 2 sec is

$\displaystyle \frac{s(2)-s(1)}{2-1} = \frac{16(2)^2- 16(1)^2}{2-1} = 48 ft/s$

However, if you were to find the velocity at the instant that $\displaystyle t = 1$ then you would calculate

$\displaystyle \lim_{\Delta t \rightarrow 0 }\; \frac{s(1+\Delta t) - s(1)}{t + \Delta t - t} = \lim_{\Delta t \rightarrow 0 }\; \frac{16(1+\Delta t)^2 - 16}{\Delta t} = 16 \lim_{\Delta t \rightarrow 0 }\; \frac{\Delta t^2 + 2 \Delta t }{\Delta t}$

and after cancelation, we obtain

$\displaystyle 16 \lim_{\Delta t \rightarrow 0 }\; \Delta t + 2 = 32 ft/s$