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**1. Homework Statement**

Prove the following theorems:

Theorem 1:

Vavg= (Vfinal+Vinitial)/2

TO Theorem 2:

Vavg=Vmp= V(t/2) <--Where V(t/2) represents a function read as V of t/2.

(t/2) is the midpoint in time for a given interval and V(t/2) is the instantaneous velocity at that time

**2. Homework Equations**

I need to prove these theorems esp theorem 2 (need to prove theorem 1 first) to relate average velocity (measured in lab) to instantaneous velocity (needed to calculate kinetic energy)

**3. The Attempt at a Solution**

I was able to prove theorem 1 using kinematic equation:

X-Xo = Vot + 1/2at^2 and V=Vo + at

Solution:

X-Xo = (2Vot)/2 + (at^2)/2

X-Xo = (2Vot + at^2)/2

(X-Xo)/t = (2Vo + at)/2

(X-Xo)/t = [(Vo + at) + Vo]/2, substitute V=Vo + at

to get,

(X-Xo)/(t-to)= (V + Vo)/2= Vavg

For theorem 2, I have to prove it using a kinematic equation too. I tried but I'm not sure if I'm doing it right.

I tried using X-Xo = 1/2 (V+Vo)t

to get,

X-Xo= [(V+Vo)t]/2

(X-Xo)/(V+Vo) = t/2

Or using V(t/2) = Vo + a(t/2) ----> V of t/2

and plug V(t/2) in X-Xo = 1/2 (V+Vo)t

X-Xo = 1/2 [{(Vo + a(t/2)} +Vo)]t

but my answer didn't make sense.