A 12.0 g bullet is fired into a 13.9kg wooden block that is hanging straight down, suspended by a 2.00 m length of light line. The bullet has a muzzle velocity of 94.2 m/s. When the bullet hits the block, the bullet stays in the block. The block swings outward so that the line it hangs from makes an angle of 4 to the vertical.
What height did the block rise to?
To determine the height to which the block rises, we need to use the principle of conservation of mechanical energy. The initial mechanical energy of the system (bullet + block) is equal to the final mechanical energy when the block is at its highest point.
1. Calculate the initial kinetic energy of the bullet:
- The mass of the bullet is given as 12.0 g, which can be converted to kilograms by dividing by 1000: 12.0 g = 0.012 kg.
- The initial velocity of the bullet is given as 94.2 m/s.
- The initial kinetic energy of the bullet is given by: KE = 0.5 * mass * velocity^2.
Substituting the given values into the formula, we get:
KE_bullet = 0.5 * 0.012 kg * (94.2 m/s)^2 = 0.0671 J (rounded to four decimal places).
2. Calculate the change in potential energy of the system:
- The mass of the block is given as 13.9 kg.
- The change in potential energy is equal to the mass of the block times the acceleration due to gravity (9.8 m/s^2) times the change in height (h).
Substituting the given values into the formula, we get:
ΔPE_system = mass_block * g * Δh.
3. Calculate the final kinetic energy of the system:
- The final kinetic energy is zero, as the bullet becomes embedded in the block and both objects move together as one.
4. Apply conservation of mechanical energy:
According to the conservation of energy principle, the initial mechanical energy (KE_bullet) is equal to the final mechanical energy (ΔPE_system + KE_system).
Thus, we have the equation:
0.0671 J = 13.9 kg * 9.8 m/s^2 * Δh + 0.5 * 13.9 kg * (v^2),
where v represents the final velocity of the block+bullet system. Since the block is at its highest point, the final velocity is zero.
5. Solve the equation for Δh:
Rearrange the equation to solve for Δh:
0.0671 J = 13.9 kg * 9.8 m/s^2 * Δh,
Δh = 0.0671 J / (13.9 kg * 9.8 m/s^2).
Calculating the value, we find:
Δh = 4.791 m (rounded to three decimal places).
Therefore, the block rises to a height of approximately 4.79 meters.
To find the height that the block rises to, we need to use the principle of conservation of momentum and the conservation of energy. Here are the steps to solve the problem:
Step 1: Find the momentum of the bullet before it hits the block.
The momentum of an object is given by the product of its mass and velocity. Since the bullet gets embedded in the block, the initial momentum of the bullet is equal to the momentum of the bullet-block system after the collision.
Momentum before collision = momentum after collision
Momentum before collision = mass of bullet × velocity of bullet
Momentum before collision = (0.012 kg) × (94.2 m/s)
Momentum before collision = 1.1304 kg·m/s
Step 2: Find the velocity of the bullet-block system after the collision.
Since the bullet stays embedded in the block, their masses combine and continue moving together as one system.
Total mass after collision = mass of bullet + mass of block
Total mass after collision = 0.012 kg + 13.9 kg
Total mass after collision = 13.912 kg
Using the principle of conservation of momentum:
Momentum after collision = total mass after collision × velocity after collision
1.1304 kg·m/s = (13.912 kg) × velocity after collision
Solving for the velocity after collision:
velocity after collision = 1.1304 kg·m/s / 13.912 kg
velocity after collision = 0.0813 m/s (approx)
Step 3: Find the height the block rises to by using the conservation of energy principle.
The initial energy of the block-bullet system is in the form of kinetic energy and gravitational potential energy. When the block rises to its highest point, all of the initial kinetic energy is converted into gravitational potential energy.
Gravitational potential energy = mass of block × gravitational acceleration × height
Initial kinetic energy = (1/2) × total mass after collision × (velocity after collision)^2
Setting the initial gravitational potential energy equal to the initial kinetic energy:
mass of block × gravitational acceleration × height = (1/2) × total mass after collision × (velocity after collision)^2
Solving for the height:
height = (0.5) × (13.912 kg) × (0.0813 m/s)^2 / (9.8 m/s^2)
height = 0.0273 m (approx)
Therefore, the block rises to a height of approximately 0.0273 meters.