Need help deriving an expression for average acceleration in terms of displacement and time

Question

I'm having trouble with this math problem.

Here's what the questions is asking:

1. Derive an expression for the average acceleration of the cart in terms of displacement and time of travel.

   • Write out the definition of average acceleration in terms of the final velocity, vf . Let $vi = 0$. You can assume the acceleration is constant.
   • Express $v_f$ in terms of the average velocity by using the fact that the average velocity equals the mean velocity $(v_f + v_i)/2$ when a = constant.
   • Finally, use the definition of average velocity $(v_{av} = Δx/Δt)$ to express the acceleration in terms of the displacement and the time of travel.

I've gotten this far:

$v_f = a_{av}(t_f-t_i)$, and $v_f = 2v_{av}-v_i$, and $v_i=0$, so

$a_{av}(t_f-t_i)=2v_{av}$, so now substitute in average velocity:

$a_{av}(t_f-t_i)=2(x_f-x_i)/(t_f-t_i)$

$a_{av}=2(x_f-x_i)/(t_f-t_i)^2$ or $a_{av}=2\Delta x / (\Delta t)^2$

Is this correct?

Answer 1

That is fine. It is dependent on the assumption (which you were given) that $v_i=0$