A 96.0-g wooden block is initially at rest on a rough horizontal surface when a 11.4-g bullet is fired horizontally into (but does not go through) it. After the impact, the block–bullet combination slides 6.5 m before coming to rest. If the coefficient of kinetic friction between block and surface is 0.750, determine the speed of the bullet immediately before impact.
To determine the speed of the bullet immediately before impact, we can use the principle of conservation of momentum.
Step 1: Calculate the initial momentum of the block-bullet system.
The formula for momentum is p = m * v, where p is momentum, m is mass, and v is velocity.
Since the block is initially at rest, its initial momentum is zero: p_block_initial = 0.
The momentum of the bullet before impact can be calculated using the given mass of the bullet and the velocity of the bullet. Let's assume the velocity of the bullet before impact is v_bullet_initial.
p_bullet_initial = m_bullet * v_bullet_initial
Step 2: Calculate the final momentum of the block-bullet system.
Since the block and bullet slide together after the impact, they have the same final velocity, which we'll call v_combined_final.
The combined mass of the block and bullet is the sum of their masses: m_combined = m_block + m_bullet.
The final momentum is given by p_combined_final = m_combined * v_combined_final.
Step 3: Apply the conservation of momentum principle.
According to the conservation of momentum principle, the initial momentum of the system is equal to the final momentum:
p_block_initial + p_bullet_initial = p_combined_final
Substituting the values we calculated:
0 + m_bullet * v_bullet_initial = (m_block + m_bullet) * v_combined_final
Step 4: Solve for the velocity of the bullet.
Rearranging the equation:
v_bullet_initial = (m_block + m_bullet) * v_combined_final / m_bullet
Substituting the given values:
v_bullet_initial = (96.0 g + 11.4 g) * v_combined_final / 11.4 g
Converting the masses to kg:
v_bullet_initial = (0.096 kg + 0.0114 kg) * v_combined_final / 0.0114 kg
v_bullet_initial = (0.1074 kg) * v_combined_final / 0.0114 kg
Step 5: Calculate the velocity of the block-bullet combination using the work done by friction.
The work done by friction is given by the formula W = F * d, where W is work, F is force, and d is distance.
The force of friction can be calculated using the coefficient of kinetic friction (μ_k) and the normal force (N). The normal force is equal to the weight of the block-bullet system: N = m_combined * g, where g is the acceleration due to gravity.
The work done by friction is also equal to the change in kinetic energy:
W = ΔKE
The change in kinetic energy is given by ΔKE = KE_final - KE_initial.
Since the block-bullet system is initially at rest, the initial kinetic energy is zero: KE_initial = 0.
The final kinetic energy can be calculated using the mass of the block-bullet system and the final velocity: KE_final = 1/2 * m_combined * v_combined_final^2.
Setting the work done by friction equal to the change in kinetic energy:
F * d = 1/2 * m_combined * v_combined_final^2
Since the force of friction is given by F = μ_k * N:
μ_k * N * d = 1/2 * m_combined * v_combined_final^2
Substituting the values:
0.750 * (m_combined * g) * 6.5 m = 1/2 * m_combined * v_combined_final^2
Simplifying:
4.875 * m_combined * g = 1/2 * m_combined * v_combined_final^2
Step 6: Solve for the velocity of the block-bullet combination.
Canceling out the mass of the block-bullet system:
4.875 * g = 1/2 * v_combined_final^2
v_combined_final^2 = 9.75 * g
v_combined_final = √(9.75 * g)
Step 7: Substitute the final velocity into the equation for v_bullet_initial.
v_bullet_initial = (0.1074 kg) * √(9.75 * g) / 0.0114 kg
v_bullet_initial = (0.1074) * √(9.75 * 9.8 m/s^2) / 0.0114
Calculating the value:
v_bullet_initial ≈ 32.988 m/s
Therefore, the speed of the bullet immediately before impact is approximately 32.988 m/s.
To determine the speed of the bullet immediately before impact, we can use the principle of conservation of momentum.
The momentum before the impact is equal to the momentum after the impact, assuming no external forces are acting on the system.
The momentum of an object is given by the product of its mass and velocity.
Let's assign variables to the known quantities:
- Mass of the wooden block: m1 = 96.0 g = 0.096 kg
- Mass of the bullet: m2 = 11.4 g = 0.0114 kg
- Initial velocity of the bullet: u2 (we need to find this)
- Final velocity of the block-bullet combination: v
- Coefficient of kinetic friction: μ = 0.750
- Distance the combination slides: d = 6.5 m
Before the impact:
- Momentum of the block: m1 * 0 (as it is at rest) = 0
- Momentum of the bullet: m2 * u2
After the impact:
- Momentum of the block-bullet combination: (m1 + m2) * v
Now we can equate the two momenta and solve for u2 (initial velocity of the bullet):
0 + m2 * u2 = (m1 + m2) * v
Simplifying the equation:
m2 * u2 = (m1 + m2) * v
Plugging in the values:
0.0114 * u2 = (0.096 + 0.0114) * v
0.0114 * u2 = 0.1074 * v
Next, we need to use the information about the friction to determine the deceleration of the block-bullet combination.
The friction force acting on the block-bullet combination can be calculated using the equation:
friction force = μ * (mass of the block-bullet combination) * gravitational acceleration
friction force = μ * (m1 + m2) * 9.8 m/s^2
Now, we can use the work-energy principle to relate the friction force to the distance the combination slides (d) and the change in kinetic energy of the system.
The work done by the friction force minus the change in kinetic energy of the system is equal to zero:
friction force * d - (1/2) * (m1 + m2) * v^2 = 0
Plugging in the values:
(μ * (m1 + m2) * 9.8) * d - (1/2) * (m1 + m2) * v^2 = 0
Now, we have two equations:
1) 0.0114 * u2 = 0.1074 * v
2) (μ * (m1 + m2) * 9.8) * d - (1/2) * (m1 + m2) * v^2 = 0
We can solve this system of equations simultaneously to find the values of u2 (initial velocity of the bullet) and v (final velocity of the block-bullet combination).