as shown in the figure a bullet of mass 0.010 kg moving horizontally suddenly strikes a block of wood of mass 1.5 kg that is suspended as a pendulum. The bullet lodges in the wood, and together they swing upward a vertical distance of 0.40 m. The length of the string is 2.0 m. What was the speed of the bullet just before it struck the wooden block?
To find the speed of the bullet just before it struck the wooden block, we can apply the concept of conservation of momentum and conservation of energy.
1. Conservation of momentum:
The total momentum before the collision is equal to the total momentum after the collision.
Initial momentum = Final momentum
The initial momentum of the bullet is given by the mass of the bullet (m1) multiplied by its initial velocity (v1).
Initial momentum = m1 * v1
After the bullet lodges in the wood, the bullet and the wood move together. The final momentum is given by the combined mass of the bullet and the wood (m1 + m2) multiplied by their final velocity (v2).
Final momentum = (m1 + m2) * v2
Since the bullet lodges in the wood, their final velocity is the same.
2. Conservation of energy:
The initial kinetic energy of the bullet is equal to the combined kinetic energy of the bullet and the wood at the highest point of their swing.
Initial kinetic energy = Final kinetic energy
The initial kinetic energy of the bullet is given by (1/2) * m1 * v1^2.
The final kinetic energy is given by (1/2) * (m1 + m2) * v2^2.
At the highest point of their swing, their velocity is zero. Therefore, the final kinetic energy is zero.
Setting the initial kinetic energy equal to zero:
(1/2) * m1 * v1^2 = 0
Simplifying, we can solve for v1:
v1^2 = 0
Taking the square root of both sides, we get:
v1 = 0
Therefore, the speed of the bullet just before it struck the wooden block is 0.
To find the speed of the bullet just before it struck the wooden block, we can use the principle of conservation of energy.
First, let's calculate the potential energy gained by the bullet and the block-wood system as they swing upward.
1. Calculate the potential energy gained by the block of wood:
- The mass of the block of wood (m_block) = 1.5 kg
- The height gained (h) = 0.40 m
- The acceleration due to gravity (g) = 9.8 m/s^2
- Potential energy gained by the block = m_block * g * h
2. Calculate the potential energy gained by the bullet:
- The mass of the bullet (m_bullet) = 0.010 kg
- The potential energy gained by the bullet is the same as the potential energy gained by the block since they are attached and move together.
3. Calculate the total potential energy gained:
- Total potential energy gained = Potential energy gained by the block + Potential energy gained by the bullet
Next, let's calculate the initial kinetic energy of the bullet before it struck the wooden block, which can be converted to potential energy when the bullet lodges in the block.
4. Calculate the mass of the combined block-wood-bullet system:
- The mass of the combined system (m_combined) = m_block + m_bullet
5. Calculate the initial kinetic energy of the bullet:
- Initial kinetic energy = (1/2) * m_combined * v_initial^2
(where v_initial is the initial velocity of the bullet before it struck the wooden block)
According to the conservation of energy principle, the initial kinetic energy of the bullet should equal the total potential energy gained by the system.
6. Equate the initial kinetic energy to the total potential energy and solve for v_initial:
Initial kinetic energy = Total potential energy gained
Now, you can use equations (4) and (6) to calculate the speed of the bullet just before it struck the wooden block.
Answer:
423m/s
Explanation:
Suppose after the impact, the bullet-block system swings upward a vertical distance of 0.4 m. That's means their kinetic energy is converted to potential energy:
where m is the total mass and h is the vertical distance traveled, v is the velocity right after the impact at, which we can solve by divide both sides my m
Let g = 9.81 m/s2
According the law of momentum conservation, momentum before and after the impact must be the same
where are the mass and velocity of the bullet before the impact, respectively. are the mass and velocity of the block before the impact, respectively, which is 0 because the block was stationary before the impact