Instantaneous velocity from avg velocity with constant accelartion

[KeilB]

Homework Statement

With constant acceleration prove that the average velocity from t₁ to t₂ =t₁ + Δt is equal to the instanous velocity in the middle of the time interval between t1 and t2.

Homework Equations

What I am looking for is a general equation that does not involve accelaration. All I have is position. By the way this is 1-D kinematics so no need for vectors.

The Attempt at a Solution

Tried all sorts of subs with Vavg= (Vi + Vf)/2 and Vavg= Δx/Δt I tried subbing these into many of the other equations such as Vf=Vi+at. I imagine there is some sort of trick to it. This thing is driving me nuts.

 

[Ninty64]

Vavg= (Vi + Vf)/2
Vf=Vi+at

It can be shown this way I believe. Here's a hint:
Vavg= (Vi + Vf)/2
Vavg= (Vi + Vf)/2 - Vi +Vi
Vavg= (Vf - Vi)/2 +Vi

 

[KeilB]

Alright still having a hard time. I got it into that form that you mentioned and was able to get Vavg=2Vf-(3at)/2 which there is a t/2 in there. Good sign I suppose but it just doesn't seem right since I am going to have to use a distance traveled to find it. I will play around with it some more but feel I'm coming to dead ends.

 

[vela]

Can you express in math what it is exactly that you're trying to show? In other word, what equation says "the average velocity from t₁ to t₂ is equal to the instantaneous velocity in the middle of the time interval between t₁ and t₂"?