find the initial velocity of a .01 kg bullet that hits a 2 kg block and the block moves .1 meters.

You need to know the force that stops the block from moving. Is it friction or is it mounted on a pendulum?

it's hanging from a string...

Did anyone mention how long a string?

Conservation of momentum gives you the initial velocity v of the block with the bullet embedded as a function of bullet speed.
Then you can get the kinetic energy of that system at the impact from (1/2) m v^2
that becomes potential energy when the system stops.
That potential energy is m g h
where h is how much it went up
but how far it went up is
L ( 1-cos theta)
where theta is the angle of the string from vertical when it stops.
and we know that .1 = L theta where theta is in radians
In other words, I need L to do the problem.

To find the initial velocity of the bullet, we can use the principle of conservation of momentum. The momentum before the collision is equal to the momentum after the collision.

The momentum of an object is given by mass times velocity (p = mv). Let's assign variables to the given quantities:
- Mass of the bullet = m1 = 0.01 kg
- Mass of the block = m2 = 2 kg
- Initial velocity of the bullet = u1 (unknown)
- Final velocity of the bullet = v1 (unknown)
- Final velocity of the block = v2 (also unknown)

According to the conservation of momentum, the total initial momentum is equal to the total final momentum. So, we can set up the equation as follows:

(m1 * u1) + (m2 * 0) = (m1 * v1) + (m2 * v2)

Since the block is initially at rest (velocity is 0), the first term on the left-hand side becomes zero.

Simplifying the equation, we get:
(m1 * u1) = (m1 * v1) + (m2 * v2)

Now, we need to use the information that the block moves 0.1 meters to find the final velocity of the block (v2).

The work done on an object can be found using the equation W = F * d, where W is work, F is force, and d is displacement. In this case, the work done on the block is equal to the change in kinetic energy.

The work done on the block would be equal to the force exerted on the block (which is provided by the bullet) multiplied by the displacement of the block. In this case, the displacement is 0.1 meters, but the force exerted by the bullet is still unknown.

We can use Newton's second law of motion to find the force exerted by the bullet on the block. According to the second law, the force is given by F = m2 * a, where F is force, m2 is the mass of the block, and a is the acceleration of the block.

Since the block moves only due to the force exerted by the bullet, the acceleration of the block is the same as the acceleration of the bullet, denoted as a1.

To find the acceleration, we can use one of the kinematic equations, which relates distance, acceleration, initial velocity, and time:
d = (u1 * t) + (1/2 * a1 * t^2)

Since the bullet hits the block and the block moves, we can assume that the time of impact is very short, causing the time (t) term to approximate to zero. Thus, we can neglect the time term, and the equation becomes:
d = (1/2 * a1 * t^2)

The equation simplifies to:
0.1 = (1/2 * a1 * 0)

Therefore, we need to further consider the motion in terms of impulse and momentum. Impulse is the force applied over time, and it is equal to the change in momentum.

The impulse experienced by the block can be written as:
Impulse = change in momentum = m2 * v2

As mentioned earlier, the momentum before the collision is equal to the momentum after the collision. The momentum of the bullet before the collision is m1 * u1, and after the collision, it is m1 * v1. Therefore, the change in momentum for the bullet is:
Change in momentum of bullet = m1 * v1 - m1 * u1

For the block, the initial momentum is 0 (as it is at rest), and the final momentum is m2 * v2. Therefore, the change in momentum for the block is:
Change in momentum of block = m2 * v2 - 0 = m2 * v2

Since the impulse experienced by the bullet and the block is the same, we can equate the two expressions for the impulse:

m1 * v1 - m1 * u1 = m2 * v2

Now, we have two equations:
1) (m1 * u1) = (m1 * v1) + (m2 * v2)
2) m1 * v1 - m1 * u1 = m2 * v2

We can solve these two equations simultaneously to find the values of u1 (initial velocity of the bullet) and v1 (final velocity of the bullet), and v2 (final velocity of the block).