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Consider the ballistic pendulum setup: A bullet of mass m with speed vo hits a pendulum bob of mass M, suspended by a massless rope of length L. The bullet gets stuck inside, making the pendulum swing up to some height. What is the maximum height the pendulum can reach? L m M Solution: At the start, the bullet has a momentum mvɔ. When they collide and the bullet gets inside, the speed of the combined masses can be obtained using conservation of linear momentum. mvo = (m + M)v, where v is the speed of the combined masses. Therefore, we get m U = т+ M At the instant the masses move, they initially have purely kinetic energy given by 1 т? T = (т+ M)u? 2 т + M This kinetic energy becomes potential energy at the maximum height. We get m2 1 (m + M)v² = (m + M)gh. %3D 2 т+ M m2 2 (т+M)2 g v0? Finally, we have h = - Question 2a m²V02 Show that the angle from the figure is given by 0 = cos-1 2g L(m+M)²

Consider the ballistic pendulum setup: A bullet of mass m with speed vo hits a pendulum bob of mass M, suspended by a massless rope of length L. The bullet gets stuck inside, making the pendulum swing up to some height. What is the maximum height the pendulum can reach? L m M Solution: At the start, the bullet has a momentum mvɔ. When they collide and the bullet gets inside, the speed of the combined masses can be obtained using conservation of linear momentum. mvo = (m + M)v, where v is the speed of the combined masses. Therefore, we get m U = т+ M At the instant the masses move, they initially have purely kinetic energy given by 1 т? T = (т+ M)u? 2 т + M This kinetic energy becomes potential energy at the maximum height. We get m2 1 (m + M)v² = (m + M)gh. %3D 2 т+ M m2 2 (т+M)2 g v0? Finally, we have h = - Question 2a m²V02 Show that the angle from the figure is given by 0 = cos-1 2g L(m+M)²

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Consider the ballistic pendulum setup: A bullet of mass m with speed vo hits a pendulum bob of mass M, suspended by a massless rope of length L. The bullet gets stuck inside, making the pendulum swing up to some height. What is the maximum height the pendulum can reach? L m M Solution: At the start, the bullet has a momentum mvn. When they collide and the bullet gets inside, the speed of the combined masses can be obtained using conservation of linear momentum. mvo = (m + M)v, where v is the speed of the combined masses. Therefore, we get m V = т+ M At the instant the masses move, they initially have purely kinetic energy given by 1 т? T = -(т + M)u? 2 т+ M This kinetic energy becomes potential energy at the maximum height. We get m2 1 (m + M)v² 1 = (m + M)gh. 2 т + M Finally, we have h = m2 v02 1 2 (m+M)2 Question 2a Show that the angle from the figure is given by 0 = cos™ ,-1 1 2g L(m+M)² m²V02
Transcribed Image Text:Consider the ballistic pendulum setup: A bullet of mass m with speed vo hits a pendulum bob of mass M, suspended by a massless rope of length L. The bullet gets stuck inside, making the pendulum swing up to some height. What is the maximum height the pendulum can reach? L m M Solution: At the start, the bullet has a momentum mvn. When they collide and the bullet gets inside, the speed of the combined masses can be obtained using conservation of linear momentum. mvo = (m + M)v, where v is the speed of the combined masses. Therefore, we get m V = т+ M At the instant the masses move, they initially have purely kinetic energy given by 1 т? T = -(т + M)u? 2 т+ M This kinetic energy becomes potential energy at the maximum height. We get m2 1 (m + M)v² 1 = (m + M)gh. 2 т + M Finally, we have h = m2 v02 1 2 (m+M)2 Question 2a Show that the angle from the figure is given by 0 = cos™ ,-1 1 2g L(m+M)² m²V02
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