Given the following formula, solve for [tex]\( v \)[/tex].

[tex]\[ s = \frac{1}{2} \alpha^2 v + c \][/tex]

A. [tex]\( v = \frac{2(s + c)}{\alpha^2} \)[/tex]

B. [tex]\( v = \frac{s + c}{2 \alpha^2} \)[/tex]

C. [tex]\( v = \frac{2(s - c)}{\alpha^2} \)[/tex]

D. [tex]\( v = \frac{s - c}{2 \alpha^2} \)[/tex]



Answer :

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]