A 27 g bullet traveling at 297 m/s is fired into a 0.468 kg wooden block anchored to a 100 N/m spring. How far is the spring compressed
To find the distance the spring is compressed, we need to use the principles of conservation of momentum and energy.
Let's break down the problem step by step:
1. Calculate the momentum of the bullet before it hits the wooden block:
Momentum = mass × velocity
Momentum of the bullet = 27 g = 0.027 kg × 297 m/s
2. As the bullet hits the wooden block, it comes to rest and transfers its momentum to the block. This results in the block gaining some velocity.
The total momentum before the collision equals the total momentum after the collision:
bullet momentum = block momentum
3. Calculate the velocity of the block after the collision:
Momentum of the bullet = Momentum of the block
0.027 kg × 297 m/s = (0.027 kg + 0.468 kg) × velocity of the block
4. Solve for the velocity of the block after the collision.
5. Now, we can calculate the kinetic energy of the block after the collision:
Kinetic energy = 0.5 × mass × velocity^2
Kinetic energy of the block = 0.5 × 0.468 kg × (velocity of the block)^2
6. The change in potential energy of the spring will be equal to the kinetic energy gained by the block:
Change in potential energy = 0.5 × spring constant × (compression)^2
Change in potential energy = 0.5 × 100 N/m × (compression)^2
7. Since the kinetic energy gained by the block is equal to the change in potential energy of the spring, we can set the two equations equal to each other and solve for the compression:
0.5 × 0.468 kg × (velocity of the block)^2 = 0.5 × 100 N/m × (compression)^2
8. Solve the equation to find the compression of the spring. Rearrange the equation and solve for the compression.
By following these steps, you will be able to find the distance the spring is compressed.