Please derive an equation using the variables provided. ?A bullet of mass m and

Question

Please derive an equation using the variables provided.

?A bullet of mass m and speed v is fired at, hits and passes completely through a pendulum bob of mass M on the end of a stiff rod of negligible mass and length L. The bullet emerges with a speed of v/2 and the pendulum bob just makes it over the top of the trajectory without falling backward in its circular path. Determine an expression for the minimum initial speed the bullet must have in order for the pendulum bob to just barely swing through a complete vertical circle. (Use any variable or symbol stated above as necessary.)

Explanation / Answer

change in KE of bullet is equal to the change in PE of pendulum

0.5m (v^2 -(v/2)^2) = Mgh

here h=2L

(v^2 -(v/2)^2) =(2 MgL)/ 0.5 m

v^2 -v^2 /4 = (2 MgL)/ 0.5 m

v^2 = (4/3) (2 MgL)/ 0.5 m

v greater than sqrt[(4/3) (2 MgL)/ 0.5 m ]

minimum value of v = sqrt[(4/3) (2 MgL)/ 0.5 m ]

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