Question
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Answer
\( v = \sqrt{\frac{2(E - M g t)}{M}} \)
Solution
1. Start with the equation:
\[
E = M g t + \frac{1}{2} M v^{2}
\]
2. Isolate the kinetic energy term by subtracting \( M g t \) from both sides:
\[
E - M g t = \frac{1}{2} M v^{2}
\]
3. Multiply both sides of the equation by 2 to eliminate the fraction:
\[
2(E - M g t) = M v^{2}
\]
4. Divide both sides by \( M \) to solve for \( v^{2} \):
\[
v^{2} = \frac{2(E - M g t)}{M}
\]
5. Take the square root of both sides to solve for \( v \):
\[
v = \sqrt{\frac{2(E - M g t)}{M}}
\]
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Simplify this solution The Deep Dive
Did you know that the formula you’ve written combines kinetic energy and gravitational potential energy into one intriguing equation? This equation can be seen in contexts like roller coasters or sports, where you can feel both the thrill of speed (kinetic energy) and the force of gravity pulling you down (potential energy). How cool is that? In real-world applications, engineers use similar principles when designing buildings, bridges, and amusement park rides. By accounting for both kinetic and potential energy, they ensure structures are safe and can handle dynamic forces, giving everyone a safe and exhilarating ride – or a sturdy place to live and work!
