A 34-g bullet traveling at 120m/s embeds itself in a wooden block on a smooth surface. The block then slides toward a spring and collides with it. The block compresses the spring (k=100 N/m) a maximum of 1.25 cm. Calculate the mass of the block of wood.
To calculate the mass of the block of wood, 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 break down the problem into different stages and calculate the values step by step:
1. Initially, the bullet is traveling with a velocity of 120 m/s. The mass of the bullet is 34 g, which is equal to 0.034 kg. The momentum of the bullet before the collision is given by the formula:
Momentum = mass x velocity
Momentum = 0.034 kg x 120 m/s = 4.08 kg·m/s
2. The bullet embeds itself in the wooden block, which means both the bullet and the wooden block move together with the same velocity after the collision. Let's assume the mass of the wooden block is 'm' kg.
Now, the total momentum after the collision is the sum of the momentum of the bullet and the momentum of the block:
Momentum = (mass of bullet + mass of block) x velocity
4.08 kg·m/s = (0.034 kg + m kg) x 120 m/s
3. The block then slides toward the spring and collides with it. The block compresses the spring by a maximum of 1.25 cm, which is equal to 0.0125 m.
According to Hooke's law, the potential energy stored in a spring is given by the formula:
Potential Energy = 0.5 x k x (compression)^2
where k is the spring constant and compression is the distance the spring is compressed.
In this case, the potential energy is equal to the kinetic energy of the block (before it hits the spring). The kinetic energy of an object is given by the formula:
Kinetic Energy = 0.5 x mass x velocity^2
So, we can set up another equation:
0.5 x k x (compression)^2 = 0.5 x (mass of block) x velocity^2
0.5 x 100 N/m x (0.0125 m)^2 = 0.5 x m kg x 120 m/s)^2
4. Solving the equation from Step 2 and substituting the value of the velocity into the equation from Step 3, we can calculate the mass of the wooden block.
4.08 kg·m/s = (0.034 kg + m kg) x 120 m/s
0.5 x 100 N/m x (0.0125 m)^2 = 0.5 x m kg x 120 m/s)^2
Solving these equations simultaneously will give us the mass of the block of wood.