Velocity-time & displacement time graphs

[CivilSigma]

I am starting to panic...

How do you find the average velocity from a Velocity-time graph. The graph has a slope of 0.75 and starts at (0,6) and ends at (8,12)

I calculated V.avg = 9. (( 6+ 12)*0.5) but the answer sheet says 12? How?

And, in general how do you find the acceleration from a position- time graph

edit: I was looking at the wrong answer sheet >.< for the first part it is 12!
but I still don't know how to find the acceleration from a position - time graph, easily.I know I am supposed to find the instantaneous velocity at several points and then construct a velocity - time graph. But is there an easier way?

 

[Chestermiller]

I am starting to panic...

How do you find the average velocity from a Velocity-time graph. The graph has a slope of 0.75 and starts at (0,6) and ends at (8,12)

I calculated V.avg = 9. (( 6+ 12)*0.5) but the answer sheet says 12? How?

And, in general how do you find the acceleration from a position- time graph

edit: I was looking at the wrong answer sheet >.< for the first part it is 12!
but I still don't know how to find the acceleration from a position - time graph, easily.I know I am supposed to find the instantaneous velocity at several points and then construct a velocity - time graph. But is there an easier way?

Yes, there is a more direct way. The acceleration is the second derivative of the velocity with respect to time. To get the acceleration directly, you need to evaluate the second derivative with respect to time numerically. A formula you can use for this is:

$$\frac{d^2x}{dt^2}=\frac{x(t+\Delta t)-2x(t)+x(t-\Delta t)}{(\Delta t)^2}$$

This is the acceleration at time t.