A bullet of mass 0.1kg is moving at a speed of 300m/s is beds in ablock of mass 4kg initially at rest on frictionless horizontal surface as shown below.the block is attached to aspring of k is 200n/m.what is the maximum compression of the spring?
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.1 * 300 = 4.1 v
v = 7.32 m/s
Ke = (1/2) m v^2 = (1/2)(4.1)(7.32)^2 = 110 Joules
110 Joules = (1/2) k x^2
To find the maximum compression of the spring, we can use the principle of conservation of momentum.
The momentum before the collision is equal to the momentum after the collision. We can calculate the initial momentum (before collision) and equate it to the final momentum (after collision).
1. Calculate the initial momentum (before collision):
Initial momentum = mass of bullet x velocity of bullet
Initial momentum = 0.1 kg x 300 m/s
Initial momentum = 30 kg m/s
2. Calculate the final momentum (after collision):
The bullet gets embedded in the block, so the mass of the bullet and the block will now move together as one system.
Final momentum = (mass of bullet + mass of block) x velocity of the system
Since the block was initially at rest, the velocity of the system will be the same as the velocity of the bullet.
Final momentum = (0.1 kg + 4 kg) x 300 m/s
Final momentum = 4.1 kg x 300 m/s
Final momentum = 1230 kg m/s
3. Apply the conservation of momentum:
Initial momentum = Final momentum
30 kg m/s = 1230 kg m/s
4. Calculate the compression of the spring:
The energy of the system is conserved. The work done by the bullet is stored as potential energy in the spring when it deforms. We can use the formula for potential energy stored in a spring:
Potential energy = (1/2) x k x compression^2
Where:
k = spring constant = 200 N/m
compression = maximum compression of the spring
Equating the initial kinetic energy of the bullet to the potential energy stored in the spring, we get:
(1/2) x mass of bullet x velocity^2 = (1/2) x k x compression^2
Substituting the values:
(1/2) x 0.1 kg x (300 m/s)^2 = (1/2) x 200 N/m x compression^2
Now, we can solve this equation for compression:
0.5 x 0.1 kg x 300^2 m^2/s^2 = 0.5 x 200 N/m x compression^2
4500 = 100 x compression^2
compression^2 = 4500/100
compression^2 = 45
compression = sqrt(45)
compression ≈ 6.71 meters
Therefore, the maximum compression of the spring is approximately 6.71 meters.