average velocity and uniform and non uniform acceleration.

Question

Consider this table, just for example.

at; t=1 v=1

at; t=2 v=3 

at; t=3 v=6

at; t=4 v=7

My doubts:

Here we can see that the acceleration isn't constant, but does it mean that it isn't constant between time interval 1-2

What i want to know is like we say that the acceleration is not constant between interval 1-2, so in examples where the acceleration is constant(like free falling body).how do we say that it is indeed constant in every interval.

 In examples of constant acceleration, we say that acceleration between a certain interval is constant, i mean in every interval it is constant, so if we assume it constant there in an interval, then why not here in non uniform acceleration for a interval i mean.

In a example of non uniform acceleration, like i have mentioned above why can't we use this formula v1+v2/2 

why can't we say that the object travelled a displacement of 2mts in the interval 1-2? in the above example, 1+3/2 will give 2m/s as the average velocity for the interval 1-2 secs so the displacement/distance travelled can be 2m/s X 1 sec= 2mts. i know this is done in examples where acceleration is constant but why can't we apply this here.why can't the acceleration be constant for intervals(some intervals) where acceleration is non uniform.

you can take your own example

thank you.

Answer 1

The constant acceleration equations will work over intervals where the acceleration is constant but when it is not constant you will need to use a Calculus approach.

In your example, just because you want the function to be constant between 1 & 2 doesn't mean it is. Here is a possible graph:

It truly is possible that the acceleration is constant between 1 & 2 but it is more reasonable that the acceleration changes gradually over the interval. Assuming the velocity is the blue curve, there is at least one point between 1 & 2 where the acceleration equals the slope of the straight line between 1 & 2.(that is the mean value theorem). But notice that the tangent line varies over that interval. If you assume that the acceleration is constant, your calculations will be close but will have some built-in error. The smaller the interval, the less the built-in error will be.