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Energy cannot be created or destroyed in any transformation. This powerful accounting principle helps us analyze everything from particle collisions, to the motion of pendulums.

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Two pendulums with respective masses of \( m_1 = 5 \text{ kg} \) and \( m_2 = 10 \text{ kg} \) hang from the same point by identical strings, as shown in the figure above. Both being initially \( h = 1 \text{ m} \) high from the lowest point, they swing down and collide elastically at the lowest point. If the velocity of mass \( m_1\) after the collision \(v_1 \) satisfies \( v_1^2=\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b?\)

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A body of mass \( m_1 = 5 \text{ kg} \) collides elastically with a stationary mass \( m_2 = 4 \text{ kg}. \) Before the collision, the velocity of mass \( m_1 \) is \( u = 2 \text{m/s} .\) After collision, masses \( m_1 \) and \( m_2 \) move in directions that make respective angles of \(\theta_1 \) and \( \theta_2 \) with the original direction that \(m_1\) had moved in. If
\( \sin{\theta_1} = \frac{4}{5}\) , what is the velocity of mass \( m_1 \) after the collision?

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Consider a one dimensional collision between two point masses with different masses \(m_1\) and \(m_2\) and initial velocities \(v_1\) and \(v_2\). We define the ratio \(r\) as the ratio of the difference in initial velocities to the difference in final velocities:

\(r=\frac{|v_{1,i}-v_{2,i}|} {|v_{1,f}-v_{2,f}|}\).

What is \(r\) if the collision is perfectly elastic?

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Two pendulums with respective masses of \( m_1 = 4 \text{ kg} \) and \( m_2 = 8 \text{ kg}\) hang from the same point by identical strings, as shown in the figure above. Both being initially \( h = 18 \text{ m} \) high from the lowest point, they swing down and collide elastically at the lowest point. What is the maximum height that pendulum \(m_1\) reaches after the collision?

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