A block of mass M is hanging from a string of length L at velocity vb1. A bullet (mass b) is shot at the block. It passes through the block with a velocity of vb2. As a result, the block swings upwards and comes to a stop at an angle of α above the starting position.
What is the initial velocity of the block in terms of L, α, and M? I don't understand how L and α are in the end answer but that's what is specified in this problem. I used momentum and energy formulas.
Energy is absorbed by the block. How much? depends on angle of swing.
Energy absorbed= Mgh=Mg(L- Lcos α) and that is equal to energy in the bullet initially minus the enrgy it leaves with.
1/2 m vb2^2=1/2 m vb1^2-MgL(1-cos α)
and from that, you can calculate vb2
To determine the initial velocity of the block in terms of L, α, and M, we can apply the principles of conservation of momentum and conservation of energy.
1. Conservation of Momentum:
The initial momentum (before the bullet strikes the block) is equal to the final momentum (after the bullet passes through the block). Since the bullet passes through the block, we can treat the bullet and the block as separate systems.
The initial momentum of the bullet is given by: Pb1 = b * vb1
The final momentum of the bullet is given by: Pb2 = b * vb2
The initial momentum of the block is zero since it starts from rest: Pb = 0
The final momentum of the block can be determined using the conservation of momentum: Pb' = M * vb', where vb' is the upward velocity of the block after the collision.
Since the bullet passes through the block, the final momentum of the system is equal to the sum of the final momentum of the bullet and the block: Pb' = Pb2 + Pb'
2. Conservation of Energy:
The initial energy of the system is equal to the final energy of the system. Initially, the only energy involved is the gravitational potential energy of the block, given by: Ep = M * g * L * (1 - cos(α))
After the collision, the energy is mainly in the kinetic energy of the block as it swings upwards. The final kinetic energy of the block is given by: Ek' = (1/2) * M * (vb')^2
Using the conservation of energy principle, we have: Ep = Ek'
Once we have the equations representing the conservation of momentum and conservation of energy, we can solve them simultaneously to find the initial velocity of the block in terms of L, α, and M.
Please note that the analysis above assumes an idealized scenario with no external forces or energy losses.