A 0.0160-kg bullet traveling horizontally at 396 m/s strikes a 3.95-kg block of wood sitting at the edge of a table. The bullet is lodged into the wood. If the table height is 1.06 m, how far from the table does the block hit the floor?

I had thought that to do this, you'd add the momentum of the two and set it equal the combined mass to find the velocity.

.0160(396)+3.95(0)=3.966(V)
V=1.59m/s

Then I set up a triangle to try to find the distance from the table and used V as the c value in the pythagorean theorem.

1.06^2+x^2=1.59^2
x=1.19m

But the book says that isn't correct. What am I doing wrong?

yes, you mixed up your units. you divided the sqrt of a velocity and a height expecting to get a height.

To find the distance from the table where the block hits the floor, we can consider the conservation of energy.

Here's how we can solve the problem:

Step 1: Calculate the initial kinetic energy of the bullet.
The initial kinetic energy of the bullet can be calculated using the formula:
Kinetic Energy = 0.5 * mass * velocity^2

Given:
Mass of the bullet (m1) = 0.0160 kg
Velocity of the bullet (v1) = 396 m/s

Initial Kinetic Energy of the bullet (KE1) = 0.5 * m1 * v1^2

Step 2: Calculate the final kinetic energy of the block-bullet system.
Since the bullet is lodged into the wood, the block-bullet system moves together after the collision. Therefore, we can calculate the final kinetic energy using the formula mentioned in step 1 with the combined mass of the block and the bullet.

Mass of the block (m2) = 3.95 kg
Combined mass of the system (m3) = m1 + m2

Final Kinetic Energy of the block-bullet system (KE2) = 0.5 * m3 * velocity^2

Step 3: Calculate the change in potential energy.
The change in potential energy can be calculated using:
Change in Potential Energy = mass * gravity * height

Given:
Height (h) = 1.06 m
Gravity (g) = 9.8 m/s^2

Change in Potential Energy (ΔPE) = m2 * g * h

Step 4: Equate the initial kinetic energy to the final kinetic energy plus the change in potential energy.
Since energy is conserved, we can equate the initial kinetic energy (KE1) to the final kinetic energy (KE2) plus the change in potential energy (ΔPE).

KE1 = KE2 + ΔPE

Step 5: Solve for velocity.
Substitute the calculated values from steps 1-3 into the equation from step 4 and solve for velocity.

Step 6: Calculate the time of flight.
The time of flight can be calculated using the formula:
Time (t) = distance (d) / velocity (v)

Given:
Height (h) = 1.06 m

Step 7: Calculate the distance from the table where the block hits the floor.
Since the initial horizontal velocity of the bullet and the block-bullet system is the same (due to the bullet being lodged into the wood), the distance traveled (d) can be calculated as the horizontal component of the initial velocity multiplied by the time of flight.

Distance (d) = v1 * t

Step 8: Substitute the values and calculate.
Substitute the calculated values from steps 1-7 into the equation from step 7 and solve for distance.

Please note that these calculations assume ideal conditions, neglecting air resistance and any effects from the table's height on the bullet-block collision.