An 8 g bullet is shot into a 4.0 kg block, at rest on a frictionless horizontal surface. The bullet remains lodged in the block. The block moves into a spring and compresses it by 9.4 cm. The spring constant is 1000 N/m. What is the initial velocity of the bullet?
Figure the energy stored in the spring. Now give that energy to KE of the block after impact. Now find the initial velocity of the block. Now set the initial momentum of the block equal to the bullet. find velocity of bullet.
To find the initial velocity of the bullet, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Let's assign variables to the given data:
Mass of the bullet, m1 = 8 g = 0.008 kg
Mass of the block, m2 = 4.0 kg
Compression of the spring, x = 9.4 cm = 0.094 m
Spring constant, k = 1000 N/m
Step 1: Calculate the initial momentum of the bullet-block system before the collision.
Momentum before the collision = mv
Momentum of the bullet before collision = m1 * velocity of the bullet before collision
Since the block is at rest, the momentum of the block before collision is zero.
Step 2: Calculate the final momentum of the bullet-block system after the collision.
Momentum after the collision = (m1 + m2) * velocity of the bullet-block system after the collision
Step 3: Apply conservation of momentum.
According to the principle of conservation of momentum:
Momentum before the collision = Momentum after the collision
m1 * velocity of the bullet before collision = (m1 + m2) * velocity of the bullet-block system after the collision
Substituting the given values,
0.008 kg * velocity of the bullet before collision = (0.008 kg + 4.0 kg) * velocity of the bullet-block system after the collision
Step 4: Calculate the final velocity of the bullet-block system after the collision.
To calculate the final velocity of the bullet-block system after the collision, we can use the concept of conservation of energy, as the compressed spring potential energy will be converted into kinetic energy of the bullet-block system.
Potential energy stored in the compressed spring = (1/2) * k * (x^2)
Kinetic energy of the bullet-block system after the collision = (1/2) * (m1 + m2) * (final velocity of the bullet-block system after the collision)^2
Setting the potential energy equal to the kinetic energy:
(1/2) * k * (x^2) = (1/2) * (m1 + m2) * (final velocity of the bullet-block system after the collision)^2
Substituting the given values,
(1/2) * 1000 N/m * (0.094 m)^2 = (1/2) * (0.008 kg + 4.0 kg) * (final velocity of the bullet-block system after the collision)^2
Step 5: Solve for the final velocity of the bullet-block system after the collision.
Momentum before the collision = Momentum after the collision equation:
0.008 kg * velocity of the bullet before collision = (0.008 kg + 4.0 kg) * velocity of the bullet-block system after the collision
0.008 kg * velocity of the bullet before collision = 4.008 kg * velocity of the bullet-block system after the collision
velocity of the bullet before collision = 4.008 kg * velocity of the bullet-block system after the collision / 0.008 kg
velocity of the bullet before collision = 501 times the velocity of the bullet-block system after the collision
Final velocity of the bullet-block system after the collision:
(1/2) * 1000 N/m * (0.094 m)^2 = (1/2) * (0.008 kg + 4.0 kg) * (final velocity of the bullet-block system after the collision)^2
44.48 J = 4.004 kg * (final velocity of the bullet-block system after the collision)^2
(final velocity of the bullet-block system after the collision)^2 = 44.48 J / 4.004 kg
(final velocity of the bullet-block system after the collision)^2 = 11.11 m^2/s^2
final velocity of the bullet-block system after the collision = sqrt(11.11 m^2/s^2)
final velocity of the bullet-block system after the collision = 3.33 m/s
velocity of the bullet before collision = 501 times the velocity of the bullet-block system after the collision
velocity of the bullet before collision = 501 * 3.33 m/s
velocity of the bullet before collision ≈ 1670 m/s
Therefore, the initial velocity of the bullet is approximately 1670 m/s.
To find the initial velocity of the bullet, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant if there are no external forces acting on it. In this case, since the block and bullet system is isolated (no external forces acting on it), the initial momentum of the bullet-block system must be equal to the final momentum.
The momentum of an object is given by the product of its mass and velocity:
Momentum = mass x velocity
Let's denote the initial velocity of the block as v_block and the initial velocity of the bullet as v_bullet. We also know the mass of the bullet (m_bullet = 8 g = 0.008 kg) and the mass of the block (m_block = 4.0 kg).
Using conservation of momentum, we can set up the equation:
m_bullet * v_bullet + m_block * v_block = (m_bullet + m_block) * v_final
Since the bullet remains lodged in the block, their final velocity will be the same. Therefore, we can simplify the equation to:
(m_bullet + m_block) * v_final = m_bullet * v_bullet + m_block * v_block
Plugging in the given values:
(0.008 kg + 4.0 kg) * v_final = 0.008 kg * v_bullet + 4.0 kg * v_block
The final velocity, v_final, can be determined from the compression of the spring. According to Hooke's Law, the potential energy stored in a spring is given by:
Potential Energy = (1/2) * k * x^2
Where k is the spring constant and x is the compression of the spring. We know the spring constant (k = 1000 N/m) and the compression of the spring (x = 9.4 cm = 0.094 m).
The potential energy stored in the spring is equal to the kinetic energy of the block and bullet system when it compresses the spring:
Potential Energy = (1/2) * m_block * v_final^2
Plugging in the values:
(1/2) * k * x^2 = (1/2) * m_block * v_final^2
Solving for v_final:
v_final^2 = (k * x^2) / m_block
v_final = sqrt((k * x^2) / m_block)
Now we have the value for v_final, we can substitute it back into the equation for conservation of momentum:
(4.008 kg) * v_final = 0.008 kg * v_bullet + 4.0 kg * v_block
Rearranging the equation to solve for v_bullet:
v_bullet = (4.008 kg * v_final - 4.0 kg * v_block) / 0.008 kg
Plugging in the values:
v_bullet = (4.008 kg * sqrt((1000 N/m * (0.094 m)^2) / 4.0 kg) - 4.0 kg * v_block) / 0.008 kg
Finally, solving this equation will give us the initial velocity of the bullet.
1/2K(x^2)=1/2(M bullet+M block)(v^2)
1/2*(1000N/m)*(0.094^2)=1/2(0.008kg+4kg)(v^2)
V=Sqrt (4.418/2.004)
V=1.48 m/s