A 1.75-kg wooden block rests on a table over a large hole as in the figure below. A 5.20-g bullet with an initial velocity vi is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of 20.0 cm.
To solve this problem, we can use the principles of conservation of momentum and conservation of mechanical energy. Let's break down the steps to find the initial velocity of the bullet:
1. Let's first find the total momentum before the collision. The momentum is the product of mass and velocity. Since the block is initially at rest, its momentum is zero.
2. The momentum of the bullet before the collision can be calculated using the formula p = mv, where p is the momentum, m is the mass, and v is the velocity. In this case, the mass of the bullet is 5.20 grams, which is equivalent to 0.00520 kg. We need the initial velocity of the bullet, so we'll denote it as vi.
3. After the collision, the bullet embeds itself in the block, so their final velocities become the same. Let's denote this final velocity as vf.
4. To find the final velocity of the bullet and the block, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision. In this case, the momentum after the collision is the momentum of the block and the bullet together, which can be calculated as (Mass of Block + Mass of Bullet) * vf.
5. Since the block and the bullet rise to a maximum height of 20.0 cm, we can use the principle of conservation of mechanical energy to find vf. The initial mechanical energy is equal to the sum of the initial potential energy (which is zero since the block is on the table) and the initial kinetic energy of the block and bullet. The final mechanical energy is equal to the sum of the final potential energy (mgh, where m is the total mass and g is the acceleration due to gravity) and the final kinetic energy (which is zero since the block and bullet momentarily stop at the maximum height).
6. Since the final kinetic energy is zero, the initial mechanical energy is equal to the final potential energy.
Now, using the information given and the equations above, we can solve for the initial velocity of the bullet (vi).
To solve this problem, we need to apply the principle of conservation of momentum and conservation of energy. Here's a step-by-step solution:
Step 1: Identify the given information:
- Mass of the wooden block (m_block) = 1.75 kg
- Mass of the bullet (m_bullet) = 5.20 g = 0.00520 kg
- Initial velocity of the bullet (v_bullet_initial) = vi (unknown)
- Maximum height reached by the block and bullet (h) = 20.0 cm = 0.20 m
Step 2: Apply the conservation of momentum principle:
According to the conservation of momentum, the total momentum before and after the collision should be equal. The momentum of the bullet before the collision is given by:
Momentum_bullet_initial = m_bullet * v_bullet_initial
After the collision, the bullet is embedded in the block, and they move together with the same velocity. Thus, the total momentum after the collision is given by:
Momentum_after_collision = (m_block + m_bullet) * v_final
Since the bullet remains in the block, the final velocity (v_final) is the same as the velocity of the block and bullet together.
Equating the two momenta:
m_bullet * v_bullet_initial = (m_block + m_bullet) * v_final
Step 3: Solve for the initial velocity of the bullet:
Rearrange the equation to find the initial velocity:
v_bullet_initial = (m_block + m_bullet) * v_final / m_bullet
Step 4: Use the conservation of energy principle:
At the maximum height, all the initial kinetic energy is converted into potential energy. We can write the conservation of energy equation as:
Initial kinetic energy = Potential energy at maximum height
The initial kinetic energy is given as:
Kinetic energy_initial = (1/2) * m_block * v_final^2
The potential energy at the maximum height is given as:
Potential energy = m_block * g * h
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 5: Solve for the final velocity:
Equate the initial kinetic energy with the potential energy and solve for v_final:
(1/2) * m_block * v_final^2 = m_block * g * h
Simplify and solve for v_final:
v_final = sqrt(2 * g * h)
Step 6: Substitute the values and calculate:
Insert the given values into the equations to find the initial velocity of the bullet:
v_final = sqrt(2 * 9.8 m/s^2 * 0.20 m)
v_final ≈ 3.14 m/s
v_bullet_initial = (1.75 kg + 0.00520 kg) * (3.14 m/s) / 0.00520 kg
Solve for v_bullet_initial to get the final answer.