A 8.3-g bullet, when fired from a gun into a 1.1-kg block of wood held in a vise, penetrates the block to a depth of 7.05cm. This block of wood is next placed on a frictionless horizontal surface, and a second 8.3-g bullet is fired from the gun into the block. To what depth (in cm) will the bullet penetrate the block in this case?
8.4
To determine the depth to which the second bullet will penetrate the block, we can use the principle of conservation of momentum.
First, let's calculate the initial velocity of the first bullet before it hits the block.
Given:
Mass of the bullet (m1) = 8.3 g = 0.0083 kg
Mass of the block (m2) = 1.1 kg
Using the principle of conservation of momentum:
m1v1 = (m1 + m2)V
Where:
v1 = initial velocity of the first bullet
V = final velocity of the block and bullet after collision
Since the block is initially at rest, the final velocity of the block and bullet after the collision (V) can be calculated as follows:
V = v1 * (m1 / (m1 + m2))
Now, let's calculate the depth to which the first bullet penetrates the block:
Given:
Penetration depth of the first bullet = 7.05 cm = 0.0705 m
The work done by the first bullet to penetrate the block is given by:
Work = Force * Distance
Since the friction force is negligible, the only force acting on the bullet is the force required to penetrate the block. This force can be calculated using:
Force = Mass * Acceleration
The acceleration can be determined using the formula:
Acceleration = (Final velocity^2 - Initial velocity^2) / (2 * Distance)
Substituting the given values, we can calculate the force needed to penetrate the block.
Now, let's move on to the second bullet and find the depth to which it will penetrate the block:
The final velocity of the block and bullet after the second collision (let's call it V') can be calculated using the same equation mentioned earlier:
V' = v2 * (m1 / (m1 + m2))
Where:
v2 = initial velocity of the second bullet
Since the block is on a frictionless surface, the kinetic energy of the system should remain constant. Therefore, the energy transfer in the first collision should be equal to the energy transfer in the second collision.
The change in kinetic energy can be calculated as follows:
Change in Kinetic Energy = (1/2) * (Total mass) * ((Final velocity)^2 - (Initial velocity)^2)
Now, we have the initial velocity (v2) and the final velocity (V') for the second bullet. We can calculate the depth to which it will penetrate the block using the formula:
Depth = (Change in Kinetic Energy) / (Force)
Substitute the values into the formula to calculate the depth to which the second bullet will penetrate the block.
Please note that while the calculations and formulas have been explained here, it is important to use the appropriate units and perform the calculations accurately to get the correct answer.