Sure, let's solve for [tex]\( u \)[/tex] step-by-step.
Given the formula:
[tex]\[ E = \frac{m}{2g} \left( v^2 - u^2 \right) \][/tex]
1. Isolate the term involving [tex]\( u \)[/tex]:
[tex]\[ E = \frac{m}{2g} \left( v^2 - u^2 \right) \][/tex]
Multiply both sides by [tex]\( \frac{2g}{m} \)[/tex] to isolate the term with [tex]\( u \)[/tex]:
[tex]\[ \frac{2gE}{m} = v^2 - u^2 \][/tex]
2. Rearrange the equation to make [tex]\( u^2 \)[/tex] the subject:
[tex]\[ u^2 = v^2 - \frac{2gE}{m} \][/tex]
3. Solve for [tex]\( u \)[/tex] by taking the square root of both sides:
[tex]\[ u = \pm \sqrt{v^2 - \frac{2gE}{m}} \][/tex]
So, the solutions for [tex]\( u \)[/tex] are:
[tex]\[ u = \sqrt{v^2 - \frac{2gE}{m}} \quad \text{or} \quad u = -\sqrt{v^2 - \frac{2gE}{m}} \][/tex]
In other words, the two possible values for [tex]\( u \)[/tex] are:
[tex]\[ u = \sqrt{v^2 - \frac{2gE}{m}} \][/tex]
[tex]\[ u = -\sqrt{v^2 - \frac{2gE}{m}} \][/tex]
