A ballistic pendulum is a device for measuring bullet speeds. One of the simplest versions consists of a block of wood hanging from two long cords. (Two cords are used so that the bottom face of the block remains parallel to the floor as the block swings upward.) A 0.013-kg bullet traveling at 280 m/s hits a 2.0-kg ballistic pendulum. However, the block is not thick enough for this bullet, and the bullet passes through the block, exiting with one-third of its original speed. How high above its original position does the block rise?

The initial kinetic energy of the bullet is given by:

KE = 0.5 * m * v^2

KE = 0.5 * 0.013 kg * (280 m/s)^2

KE = 15.84 J

The final kinetic energy of the bullet is given by:

KE = 0.5 * m * v^2

KE = 0.5 * 0.013 kg * (93.3 m/s)^2

KE = 4.35 J

The change in kinetic energy of the bullet is given by:

ΔKE = KEf - KEi

ΔKE = 4.35 J - 15.84 J

ΔKE = -11.49 J

The change in kinetic energy of the bullet is equal to the change in potential energy of the block. The change in potential energy of the block is given by:

ΔPE = m * g * h

Where m is the mass of the block, g is the acceleration due to gravity, and h is the height the block rises.

We can rearrange this equation to solve for h:

h = ΔPE / (m * g)

h = (-11.49 J) / (2.0 kg * 9.8 m/s^2)

h = -0.58 m

The block rises 0.58 m above its original position.

To solve this problem, we can start by using the principle of conservation of linear momentum. According to this principle, the total linear momentum before the collision is equal to the total linear momentum after the collision.

Here's how we can solve it step-by-step:

Step 1: Calculate the momentum before the collision
The momentum of the bullet before the collision can be calculated using the formula:

P_initial = m * v_initial

Where:
m = mass of the bullet = 0.013 kg
v_initial = initial velocity of the bullet = 280 m/s

P_initial = 0.013 kg * 280 m/s

Step 2: Calculate the momentum after the collision
Since the bullet exits the block with one-third of its original speed, the velocity after the collision can be calculated as:

v_final = (1/3) * v_initial

The total mass after the collision remains the same as the mass of the bullet, so the momentum after the collision can be calculated as:

P_final = m * v_final

P_final = 0.013 kg * (1/3) * 280 m/s

Step 3: Apply the conservation of linear momentum
According to the conservation of linear momentum, the total momentum before the collision should be equal to the total momentum after the collision.

P_initial = P_final

0.013 kg * 280 m/s = 0.013 kg * (1/3) * 280 m/s

Step 4: Solve for v_final
We can solve the equation to find the velocity after the collision:

0.013 kg * 280 m/s = 0.013 kg * (1/3) * v_final

Step 5: Calculate the height the block rises

The change in the kinetic energy of the bullet is equal to the work done on the block, which is given by the gravitational potential energy:

ΔKE = m * g * h

Where:
m = mass of the block = 2.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height above the initial position

To find the height, we can rearrange the formula:

h = ΔKE / (m * g)

h = [(1/2) * m * v_final^2] / (m * g)

Substituting the values:

h = (1/2) * (1/3) * (280 m/s)^2 / (2.0 kg * 9.8 m/s^2)

Solving this equation will give us the height above the initial position that the block rises.

Please note that this calculation assumes no energy loss due to friction or air resistance.

To find the height above its original position that the block rises after being hit by the bullet, we can apply the principle of conservation of momentum.

First, let's calculate the momentum of the bullet before it hits the block.
Momentum (p) is calculated by multiplying the mass (m) by the velocity (v).
So, momentum of the bullet before collision = mass of the bullet × velocity of the bullet
Momentum before collision = (0.013 kg) × (280 m/s)

Next, let's calculate the momentum of the bullet after it goes through the block.
Given that the bullet exits the block with one-third of its original speed, we can calculate its new velocity.
New velocity of the bullet = original velocity of the bullet / 3
New velocity of the bullet = 280 m/s / 3

Now, let's calculate the momentum of the bullet after the collision.
Momentum after collision = mass of the bullet × velocity of the bullet after collision
Momentum after collision = (0.013 kg) × (280 m/s / 3)

According to the principle of conservation of momentum, the momentum before the collision should equal the momentum after the collision.
So, the momentum before collision = momentum after collision
(0.013 kg) × (280 m/s) = (0.013 kg) × (280 m/s / 3) + (mass of the pendulum + mass of the bullet) × final velocity of the block

Since the bullet passes through the block, the final velocity of the block is zero (as it momentarily comes to rest).
Therefore, we can simplify the equation to solve for the combined mass of the pendulum and the bullet.

(0.013 kg) × (280 m/s) = (0.013 kg) × (280 m/s / 3) + (mass of the pendulum + 0.013 kg) × 0

Simplifying further:
0.013 kg × 280 m/s = 0.013 kg × (280 m/s / 3)
9.8 kg⋅m/s = 0.013 kg × 93.3 m/s

Now, we can solve for the mass of the pendulum plus the mass of the bullet:
mass of the pendulum + 0.013 kg = 9.8 kg

Solving for the mass of the pendulum:
mass of the pendulum = 9.8 kg - 0.013 kg
mass of the pendulum = 9.787 kg

Now that we have the mass of the pendulum, we can find the height it rises above its original position using the conservation of mechanical energy.

The potential energy gained by the block is equal to the initial kinetic energy of the bullet:
Potential energy gained by the block = Kinetic energy of the bullet before the collision
Potential energy (mgh) = (1/2) × mass of the bullet × (initial velocity of the bullet)^2

Finally, we can rearrange the equation and solve for the height (h):

h = [mass of the bullet × (initial velocity of the bullet)^2] / [2 × (mass of the pendulum × gravity)]

Plugging in the values:
h = [(0.013 kg) × (280 m/s)^2] / [2 × (9.787 kg × 9.8 m/s^2)]

Now calculate the value to find the height above its original position that the block rises.