To solve for [tex]\(v\)[/tex] in the given equation
[tex]\[ s = \frac{1}{2} a^2 v + c, \][/tex]
we need to isolate [tex]\(v\)[/tex] on one side of the equation. Here is the step-by-step process:
1. Start with the given equation:
[tex]\[ s = \frac{1}{2} a^2 v + c. \][/tex]
2. Subtract [tex]\(c\)[/tex] from both sides to account for the constant on the right-hand side:
[tex]\[ s - c = \frac{1}{2} a^2 v. \][/tex]
3. To isolate [tex]\(v\)[/tex], we need to eliminate the coefficient [tex]\(\frac{1}{2} a^2\)[/tex] in front of [tex]\(v\)[/tex]. Do this by multiplying both sides of the equation by [tex]\(\frac{2}{a^2}\)[/tex] (the reciprocal of [tex]\(\frac{1}{2} a^2\)[/tex]):
[tex]\[ \frac{2}{a^2} (s - c) = v. \][/tex]
4. Simplify the right-hand side:
[tex]\[ v = \frac{2(s-c)}{a^2}. \][/tex]
Next, we can verify which of the given choices matches our derived solution:
A. [tex]\( v = \frac{2(s+c)}{a^2} \)[/tex]
B. [tex]\( v = \frac{(s+c)}{2a^2} \)[/tex]
C. [tex]\( v = \frac{2(s-c)}{a^2} \)[/tex]
D. [tex]\( v = \frac{(s-c)}{2a^2} \)[/tex]
Comparing our derived solution [tex]\( v = \frac{2(s-c)}{a^2} \)[/tex] with the choices:
- Option C: [tex]\( v = \frac{2(s-c)}{a^2} \)[/tex] is identical to our solution.
Thus, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
