A 5.0-g bullet traveling horizontally at an unknown speed hits and embeds itself in a 0.195-kg block resting on a frictionless table. The block slides into and compresses a 180-N/m spring a distance of 5.0×10^−2 m before stopping the block and bullet.
To find the unknown speed of the bullet before it hits the block, we can use the principle of conservation of momentum. This principle states that the total momentum before a collision is equal to the total momentum after the collision, assuming no external forces are acting on the system.
First, let's define the variables:
- m₁ = mass of the bullet = 5.0 g = 5.0 × 10^(-3) kg
- v₁ = initial velocity of the bullet (unknown)
- m₂ = mass of the block = 0.195 kg
- v₂ = velocity of the block and bullet after the collision (final velocity)
According to the conservation of momentum principle, we have:
m₁ * v₁ + m₂ * 0 = (m₁ + m₂) * v₂
Since the bullet embeds itself into the block, they will have the same final velocity after the collision. Let's call this final velocity v.
m₁ * v₁ + m₂ * 0 = (m₁ + m₂) * v
Simplifying further:
m₁ * v₁ = (m₁ + m₂) * v
Now we can substitute the given values:
(5.0 × 10^(-3) kg) * v₁ = (5.0 × 10^(-3) kg + 0.195 kg) * v
Solving for v₁:
v₁ = [(5.0 × 10^(-3) kg + 0.195 kg) * v] / (5.0 × 10^(-3) kg)
Now we need to find the final velocity v. To do this, we will use the principle of conservation of mechanical energy. The work done by the spring is equal to the change in potential energy of the block.
Based on Hooke's Law, the potential energy stored in a spring can be calculated using the formula:
PE = (1/2) * k * x²
Where:
- PE is the potential energy stored in the spring
- k is the stiffness constant of the spring (180 N/m)
- x is the distance the spring is compressed (5.0 × 10^(-2) m)
The work done by the spring can be calculated using the formula:
Work = PE_f - PE_i
Since the block and bullet are initially at rest, the initial potential energy PE_i is zero.
Work = PE_f
Therefore:
Work = (1/2) * k * x²
We know that the work done by the spring is also equal to the change in kinetic energy of the block-bullet system. The change in kinetic energy ΔKE can be calculated using the formula:
ΔKE = (1/2) * (m₁ + m₂) * v²
Setting the work done by the spring equal to the change in kinetic energy:
(1/2) * k * x² = (1/2) * (m₁ + m₂) * v²
Now we can substitute the given values:
(1/2) * (180 N/m) * (5.0 × 10^(-2) m)² = (1/2) * (5.0 × 10^(-3) kg + 0.195 kg) * v²
Solving for v:
v = √[(2 * (1/2) * (180 N/m) * (5.0 × 10^(-2) m)²) / (5.0 × 10^(-3) kg + 0.195 kg)]
Finally, substitute this value of v into the equation for v₁:
v₁ = [(5.0 × 10^(-3) kg + 0.195 kg) * v] / (5.0 × 10^(-3) kg)
Now, plug in the values and calculate v₁ to find the unknown speed of the bullet before it hits the block.
To find the unknown speed of the bullet, we can use the principle of conservation of momentum. The momentum before the collision is equal to the momentum after the collision.
Step 1: Find the initial momentum before the collision.
The initial momentum of the bullet can be calculated using the formula:
P_initial = m_bullet * v_bullet
where:
m_bullet = mass of the bullet = 5.0 g = 0.005 kg (converted from grams to kilograms)
v_bullet = unknown speed of the bullet
P_initial = 0.005 kg * v_bullet
Step 2: Find the final momentum after the collision.
After the collision, the bullet is embedded in the block, so the final momentum is given by:
P_final = (m_bullet + m_block) * v'
where:
m_block = mass of the block = 0.195 kg
v' = final velocity of the block and bullet system
P_final = (0.005 kg + 0.195 kg) * v'
Step 3: Apply the principle of conservation of momentum.
According to the principle of conservation of momentum, the initial momentum is equal to the final momentum:
P_initial = P_final
0.005 kg * v_bullet = (0.005 kg + 0.195 kg) * v'
Simplifying the equation:
0.005 kg * v_bullet = 0.200 kg * v'
Step 4: Calculate the unknown velocity.
To find the unknown velocity v', we need to know the compression distance of the spring.
Given:
k = 180 N/m (spring constant)
x = 5.0 × 10^-2 m (compression distance)
The potential energy stored in the compressed spring can be calculated using the formula:
Potential energy = (1/2) k x^2
Step 5: Find the potential energy stored in the spring.
Potential energy = (1/2) * 180 N/m * (5.0 × 10^-2 m)^2
Step 6: Equate the potential energy to the kinetic energy of the block and bullet system.
According to the principle of conservation of mechanical energy, the potential energy stored in the spring is equal to the kinetic energy of the block and bullet system:
Potential energy = Kinetic energy
(1/2) * m_block * v'^2 = (1/2) * (m_bullet + m_block) * v'^2
Substituting the known values:
(1/2) * 0.195 kg * v'^2 = (1/2) * (0.005 kg + 0.195 kg) * v'^2
Step 7: Solve for the unknown velocity v'.
Simplifying the equation:
0.195 kg * v'^2 = 0.200 kg * v'^2
Step 8: Cancel out the squared terms and solve for v'.
Subtracting 0.200 kg * v'^2 from both sides of the equation:
0.195 kg * v'^2 - 0.200 kg * v'^2 = 0
0.005 kg * v'^2 = 0
Since the left side of the equation is equal to zero, it implies that v' should also be zero.
Therefore, the unknown velocity v' is zero, which means that the block and bullet system comes to a complete stop after the collision.
Note: This result indicates that the block and bullet system loses all its initial energy, converting it into potential energy stored in the compressed spring.
energy in the block and bullet=spring energy
1/2 (.005+.195)V=1/2 180*(5E-2)^2
solve for V, the velocity of the block with bullet.
Now consider the bullet and block impact, conservation of momentum applies.
.05Vb=(.195+.005)Vabove
solve for the velocity of the bullet, Vb