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Example 1: Say an object moves with velocity 3t in the time t=0 till t=2. Find distance covered.

Initial velocity = 0.

Final velocity = 6 disp. unit/ time unit.

Avg. Velocity = 3 disp. unit/ time unit.

Distance covered = Avg. velocity x time = 6 disp. units.

Using s = ut +1/2 a[tex]t^{2}[/tex] we get 6 again. Amazing!

Example 2: Force acting on a box of mass 1 unit is 3t in the time t=0 till t=2. Find work done by the Force. Box is initially at rest to your frame.

No other forces act on it.

Initial force = 0

Final force = 6 units.

Avg. force = 3 units.

Now avg. accn. = 3 units [mass = 1]

As in previous sum, displacement = 6 units.

Work done = 3 x 6 = 18 units. This comes out fine if you work it out the normal way also.

Now onto my questions.

If you noticed both were linear variations. How do I find the average of any polynomial function? I would find that VERY useful. For instance I found out for a cos/sin function average is 1/[tex]\sqrt{2}[/tex] of the co-efficient of the cos function. Isn't that fantastic?

Also one more. I was given a problem that the charge density of a sphere varies as [tex]\beta[/tex]t. But when I tried average, it doesn't work although it seems to be a linear variation.

Why doesn't it work?