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.
Well, well, well, looks like this bullet really made an impact! Let's dive into the world of physics and solve this puzzle.
First things first, let's figure out the initial velocity of the wooden block. Since it's initially at rest, we can say its initial velocity is zero.
Now, let's calculate the momentum before the impact. The momentum of the bullet will be its mass multiplied by its velocity, and since we know the mass of the bullet is 11.4 grams, we need to convert it to kilograms. So, the mass of the bullet is 0.0114 kg.
Since the bullet is fired horizontally into the block, the horizontal component of the bullet's velocity will be the same as the velocity of the block after impact.
The momentum before the impact is equal to the momentum after the impact, so we can say:
(mass of bullet) × (velocity of bullet) = (mass of block) × (velocity of block)
Plugging in the values we know, we get:
0.0114 kg × (velocity of bullet) = 0.096 kg × (velocity of block)
Simplifying the equation, we have:
(velocity of bullet) = (0.096 kg × (velocity of block)) / 0.0114 kg
Now, let's find the velocity of the block after the impact. We have the distance it traveled, which is 6.5 m, and we can use the work-energy theorem to find it. The work done on the block is equal to the force of friction multiplied by the distance it traveled. The work done is given by W = ΔK = 0.5 × (mass of block) × (velocity of block)^2, where ΔK is the change in kinetic energy.
So, the work done is:
W = (force of friction) × (distance) = (coefficient of kinetic friction) × (normal force) × (distance)
The normal force is equal to the weight of the block, which is (mass of block) × (acceleration due to gravity). So we have:
W = (coefficient of kinetic friction) × (mass of block) × (acceleration due to gravity) × (distance)
Now, since the work done is equal to the change in kinetic energy, we can equate these two equations:
0.5 × (mass of block) × (velocity of block)^2 = (coefficient of kinetic friction) × (mass of block) × (acceleration due to gravity) × (distance)
We can simplify this equation to:
0.5 × (velocity of block)^2 = (coefficient of kinetic friction) × (acceleration due to gravity) × (distance)
Now, we can rearrange the equation to solve for the velocity of the block:
(velocity of block)^2 = (2 × (coefficient of kinetic friction) × (acceleration due to gravity) × (distance))
Taking the square root of both sides, we have:
velocity of block = √(2 × (coefficient of kinetic friction) × (acceleration due to gravity) × (distance))
Plugging in the given values, we have:
velocity of block = √(2 × 0.750 × 9.8 m/s^2 × 6.5 m)
Finally, the velocity of the bullet immediately before impact is equal to the velocity of the block after the impact, which we calculated earlier.
So, the speed of the bullet immediately before impact is approximately equal to the square root of the velocity of the block after the impact.
To determine the speed of the bullet immediately before impact, we can apply the principle of conservation of momentum.
1. Calculate the initial momentum of the bullet and the block:
Momentum = mass × velocity
The mass of the bullet is given as 11.4 g, which we convert to kg:
mass_bullet = 11.4 g = 11.4 g ÷ 1000 g/kg = 0.0114 kg
The mass of the block is given as 96.0 g, which we convert to kg as well:
mass_block = 96.0 g = 96.0 g ÷ 1000 g/kg = 0.0960 kg
Since the block is at rest initially, its initial velocity is 0 m/s:
velocity_block_initial = 0 m/s
Therefore, the initial momentum of the system is:
initial_momentum = mass_bullet × velocity_bullet_initial + mass_block × velocity_block_initial
Since the block is at rest initially, the equation simplifies to:
initial_momentum = mass_bullet × velocity_bullet_initial
2. Calculate the final momentum of the block-bullet combination after they slide 6.5 m:
According to the law of conservation of momentum, the total momentum before and after the impact is the same.
The final velocity of the block-bullet combination is 0 m/s, as it comes to rest.
Now, the momentum after the impact is:
final_momentum = mass_block-bullet × velocity_block-bullet_final
3. Find the speed of the bullet immediately before impact:
By equating the initial and final momentum, we can solve for the velocity_bullet_initial:
initial_momentum = final_momentum
mass_bullet × velocity_bullet_initial = mass_block-bullet × velocity_block-bullet_final
Rearranging the equation:
velocity_bullet_initial = (mass_block-bullet × velocity_block-bullet_final) / mass_bullet
We can calculate the velocity_block-bullet_final using the work-energy theorem:
The work done by friction is equal to the initial kinetic energy of the block-bullet combination. The work done by friction can be calculated as the product of the frictional force (frictional force = coefficient of kinetic friction × normal force) and the distance over which the block-bullet combination slides.
Since the block is at rest initially, the normal force is equal to the weight of the block, which can be calculated as the product of mass and acceleration due to gravity (F_gravity = mass × g).
The work done by friction can be calculated as:
work_friction = frictional force × distance
kinetic_energy_initial = work_friction
Kinetic energy_initial is given as:
kinetic_energy_initial = 0.5 × mass_block-bullet × (velocity_block-bullet_final)^2
Substitute the velocity_block-bullet_final:
kinetic_energy_initial = 0.5 × mass_block-bullet × (velocity_block-bullet_final)^2
Now, solve for velocity_bullet_initial using the derived equation from momentum conservation.
Finally, substitute the given values:
mass_bullet = 0.0114 kg
mass_block = 0.0960 kg
velocity_block-bullet_final = 0 m/s
distance = 6.5 m
coefficient of kinetic friction = 0.750
Calculate velocity_bullet_initial.
To determine the speed of the bullet immediately before impact, we need to use the principle of conservation of momentum, which states that the total momentum of an isolated system remains constant before and after the interaction. In this case, the system consists of the wooden block and the bullet.
Let's break down the problem step by step:
Step 1: Calculate the initial momentum of the block-bullet system
Since the wooden block is initially at rest, its initial momentum is zero.
The initial momentum of the bullet is given by:
p_bullet_initial = m_bullet * v_bullet
where m_bullet is the mass of the bullet (11.4 g), and v_bullet is the speed of the bullet just before impact (what we're trying to find).
Step 2: Calculate the final momentum of the block-bullet system
After the impact, the block-bullet combination moves together with a common final velocity (v_final) before coming to rest.
The final momentum of the system is given by:
p_final = (m_block + m_bullet) * v_final
where m_block is the mass of the wooden block (96.0 g) and m_bullet is the mass of the bullet (11.4 g).
Step 3: Apply the principle of conservation of momentum
According to the principle of conservation of momentum, the initial momentum of the system (zero) is equal to the final momentum of the system:
0 = (m_block + m_bullet) * v_final - m_bullet * v_bullet
Step 4: Solve for v_bullet
Rearranging the equation, we get:
(m_block + m_bullet) * v_final = m_bullet * v_bullet
Now, we need to determine the final velocity of the system (v_final) before it comes to rest. We can use the equation of motion for a block sliding on a rough horizontal surface:
v_final^2 = v_initial^2 - 2 * μ * g * d
where v_initial is the initial velocity of the system (which is the velocity of the bullet just before impact), μ is the coefficient of kinetic friction (0.750), g is the acceleration due to gravity (9.8 m/s^2), and d is the distance the system slides before coming to rest (6.5 m).
Step 5: Calculate v_final using the equation of motion
Since the system comes to rest, the final velocity is zero. Plugging in the given values, we have:
0 = v_bullet^2 - 2 * 0.750 * 9.8 * 6.5
Step 6: Solve for v_bullet
Simplifying the equation, we get:
v_bullet^2 = 98.7
Step 7: Calculate v_bullet
Taking the square root of both sides, we find:
v_bullet = √98.7 ≈ 9.934 m/s
Therefore, the speed of the bullet immediately before impact is approximately 9.934 m/s