1. ## Rectilinear Motion

Can someone explain what average velocity is? I know that it's change in position/change in time = $\frac{\Delta s}{\Delta t}=\frac{f(t+\Delta t)-f(t)}{\Delta t}=v_\text{avg}$, 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 = $\frac{\Delta s}{\Delta t}=\frac{f(t+\Delta t)-f(t)}{\Delta t}=v_\text{avg}$, 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 $\Delta t$. For average velocity, $\Delta t$ is some fixed amount, like $\Delta t = 1$ or $\Delta t = 5$ whereas for instanteous velocity, $\Delta t \rightarrow 0$. For example, suppose you are given the postion function.

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

Then the average velocity going from 1 to 2 sec is

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

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

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

and after cancelation, we obtain

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

3. let's say you take a trip over the river and through the woods to grandmother's house, covering a distance of 150 miles in 3 hours.

your average velocity for the entire trip is (150 miles)/(3 hrs) = 50 mph

now, if you traveled via car, does that mean your speedometer always pointed at 50 mph? ... of course not. The speedometer reads the instantaneous speed, or the magnitude of velocity, at every instant of time.

as the measured time interval gets smaller, the average velocity over that measured time period approaches the instantaneous velocity at a specific time within that interval of time.

see the difference?