Assume that before the interaction the potential energy of the block is 0. After the dart sticks to the block, the block also acquires the kinetic energy and they move as one, meanwhile the kinetic energy converts to the potential energy and some other kinetic energy necessary to move along the circle:
According to the condition, the block and the dart must go up for a height of 2R, that is, acquire the potential energy of
According to law of conservation of energy (just substitute everything):
where "v_r" - speed required to "neutralize" the acceleration due to gravity and provide the circular motion:
Therefore:
How do we find the speed of the dart v? Remember about momentum conservation:
Substitute:
"u=\\frac{m+M}{m}\\sqrt{5gR}=11\\text{ m\/s}."
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