A bullet (m = 0.0290 kg) is fired with a speed of 94.00 m/s and hits a block (M = 2.70 kg) supported by two light strings as shown, stopping quickly. Find the height to which the block rises

problem 1 is conservation of momentum

m (v) + M(0) = (M+m)V

problem 2 is conservation of energy
Ke after collision = (1/2)(m+M)V^2
which = (M+m)g h

To find the height to which the block rises, we can use the principle of conservation of momentum and conservation of energy.

1. Conservation of momentum:
Before the collision, the bullet and the block are separate. After the collision, they stick together and move as one object. The momentum before the collision is equal to the momentum after the collision.

The momentum before the collision (bullet) is given by:
P_initial = m_bullet * v_bullet

The momentum after the collision (bullet + block) is given by:
P_final = (m_bullet + M) * v_final

Since the bullet stops after the collision, the final velocity (v_final) is equal to 0 m/s.

Setting the initial momentum equal to the final momentum:
m_bullet * v_bullet = (m_bullet + M) * 0

2. Conservation of energy:
The kinetic energy of the bullet before the collision is equal to the potential energy of the block after it rises.

The kinetic energy before the collision (bullet) is given by:
KE_initial = (1/2) * m_bullet * v_bullet^2

The potential energy of the block after it rises is given by:
PE_final = M * g * h

where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the height to be determined.

Setting the initial kinetic energy equal to the final potential energy:
(1/2) * m_bullet * v_bullet^2 = M * g * h

Now we can solve these equations to find the height (h) to which the block rises.

First, let's calculate the mass of the bullet (m_bullet) using its given value (m = 0.0290 kg).

Next, calculate the initial momentum (P_initial) using the given bullet speed (v_bullet = 94.00 m/s).

Then, calculate the height (h) using the mass of the block (M = 2.70 kg) and gravitational acceleration (g = 9.8 m/s^2). Substitute these values into the equation (1/2) * m_bullet * v_bullet^2 = M * g * h and solve for h.