A bullet with mass 4.91 g is fired horizontally into a 1.981-kg block attached to a horizontal spring. The spring has a constant 6.25 102 N/m and reaches a maximum compression of 6.38 cm.
1) find the initial speed of the bullet-block system
2) find the speed of the bullet
To solve this problem, we need to consider the conservation of momentum and the spring force.
1. First, let's calculate the initial velocity of the bullet before it hits the block.
- The mass of the bullet is given as 4.91 g, which is equal to 0.00491 kg.
- The momentum is given by the equation P = m * v, where P is momentum, m is mass, and v is velocity.
- Since the bullet is fired horizontally, it has no vertical velocity and its initial momentum is zero.
- Therefore, the initial momentum of the system is zero, which means the momentum of the bullet is equal in magnitude but opposite in direction to the momentum of the block after impact.
2. Next, let's calculate the final velocity of the bullet and block after the impact.
- We can use the conservation of momentum, which states that the initial momentum is equal to the final momentum.
- Since the bullet sticks to the block after impact, their masses will be considered together.
- The final momentum is given by: (m_bullet + m_block) * v_final.
- We set the final momentum equal to zero to account for the momentum being zero after the bullet comes to rest.
- Therefore, (0.00491 kg + 1.981 kg) * v_final = 0.
- Solving for v_final, we have v_final = 0 m/s.
3. Now, let's calculate the compression of the spring using the spring force equation.
- The spring constant is given as 6.25 * 10^2 N/m, and the maximum compression is 6.38 cm, or 0.0638 m.
- The restoring force of the spring is given by: F = k * x, where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.
- The spring force is equal in magnitude but opposite in direction to the force exerted by the block on the spring.
- Therefore, we can set the spring force equal to the force exerted by the block on the spring, which is given by Newton's second law, F = m_block * a, where a is acceleration.
- We can find the acceleration by using Hooke's law, F = k * x = m_block * a.
- Rearranging the equation, we have a = (k * x) / m_block.
- Plugging in the values, we get a = (6.25 * 10^2 N/m) * (0.0638 m) / 1.981 kg.
- Calculating the expression, we find that a ≈ 2.014 m/s^2.
To summarize:
- The initial velocity of the bullet is zero.
- The final velocity of the bullet and block combined is zero.
- The compression of the spring is approximately 2.014 m/s^2.