To solve the problem, we need to derive the formula for $a$ (acceleration) in terms of $v$, $d$, and $t$. Let’s analyze the given equation step by step:

The equation for distance is given as: $d = vt - \frac{1}{2} a t^2$

Rearrange the equation to isolate $a$: $\frac{1}{2} a t^2 = vt - d$ $a t^2 = 2(vt - d)$ $a = \frac{2(vt - d)}{t^2}$

Now, simplify the formula: $a = 2 \left( \frac{v}{t} - \frac{d}{t^2} \right)$

Thus, the correct answer is: $\boxed{\text{C) } a = 2 \left( \frac{v}{t} - \frac{d}{t^2} \right)}$

Let me know if you'd like further explanation or have any questions!

5 Related Questions:

1. How does the equation for uniformly accelerated motion relate to this derivation?
2. Can this equation be applied when $t = 0$? Why or why not?
3. How would the formula change if $v$ was the initial velocity instead of final velocity?
4. What are the physical units of $a$, $v$, $d$, and $t$, and how do they relate in this equation?
5. How can you use this formula in real-life scenarios like calculating braking distances?

Tip:

Always ensure units are consistent when solving physics problems. For example, time should always be in seconds, and distances in meters, to maintain SI units.