A 16.0 g bullet is fired horizontally into a 87 g wooden block that is initially at rest on a frictionless horizontal surface and connected to a spring having spring constant 168 N/m. The bullet becomes embedded in the block. The bullet block system compresses the spring by a maximum of 1.30 cm. (a) What is the velocity of the block once the bullet is imbedded?

To find the velocity of the block once the bullet is embedded, we can use the principle of conservation of momentum.

1. First, let's calculate the initial momentum of the system before the bullet is embedded. The initial momentum is the product of the bullet's mass (m1) and its initial velocity (v1). Since the bullet is fired horizontally, its initial velocity is constant and can be assumed as known.

Momentum before = m1 * v1

2. Next, let's calculate the final momentum of the system after the bullet is embedded. Since the block and the bullet are now moving together, their combined mass (m2) will be the sum of the mass of the bullet and the mass of the block.

m2 = m1 + m(block)

The final velocity (v2) of the block and bullet combination can be calculated by dividing the momentum after the collision by the combined mass (m2).

Momentum after = m2 * v2
v2 = Momentum after / m2

3. Now, let's calculate the momentum after the collision. The momentum is the product of the combined mass (m2) and the final velocity (v2).

Momentum after = m2 * v2

4. We can also assume that during the collision process, no external forces are acting on the bullet-block system. Therefore, we can apply the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.

Momentum before = Momentum after

Now we have all the necessary equations to solve for v2, the velocity of the block once the bullet is embedded.

Let's plug in the given values:
m1 = 16.0 g = 0.016 kg (mass of the bullet)
v1 = unknown (initial velocity of the bullet)
m(block) = 87 g = 0.087 kg (mass of the block)
v2 = unknown (final velocity of the bullet and block combination)
m2 = m1 + m(block) = 0.016 kg + 0.087 kg

Using the principle of conservation of momentum, we can set up the equation:

m1 * v1 = m2 * v2

Solving for v2:

v2 = (m1 * v1) / m2

However, we still don't know the value of v1, so it cannot be directly calculated. Additional information is required to determine the initial velocity of the bullet.