A 25 gm bullet is fired into a 2.25kg ballistic pendulum and becomes embedded in it. If the pendulum moves a vertical distance of 10cm, calculate:
A. The speed of the system after collision
B. The initial speed of the bullet
C. The percentage of the energy lost in the process.
To solve this problem, we need to apply the principle of conservation of momentum and conservation of energy.
A. The speed of the system after the collision:
1. First, we need to determine the velocity of the pendulum immediately after the bullet is embedded in it. Assuming no external forces act on the system, we can use the principle of conservation of momentum.
- The initial momentum of the bullet is given by: p_initial = m_bullet * v_bullet, where m_bullet is the mass of the bullet and v_bullet is the initial velocity of the bullet.
- The final momentum of the bullet and pendulum together is given by: p_final = (m_bullet + m_pendulum) * v_final, where m_pendulum is the mass of the pendulum, and v_final is the final velocity of the bullet and pendulum together.
- According to the principle of conservation of momentum, the initial momentum is equal to the final momentum. Therefore, we have: p_initial = p_final.
2. Now, substitute the values into the equation: m_bullet * v_bullet = (m_bullet + m_pendulum) * v_final.
3. Rearrange the equation and solve for v_final: v_final = (m_bullet * v_bullet) / (m_bullet + m_pendulum).
- Here, the values are given as m_bullet = 25 gm = 0.025 kg, m_pendulum = 2.25 kg, and v_bullet (which we need to calculate in the next step).
4. Calculate the value of v_final using the given values.
B. The initial speed of the bullet:
- To calculate the initial speed of the bullet, we need to find the value of v_bullet.
- Since the bullet is embedded in the ballistic pendulum, the final velocity of the bullet is zero (v_final = 0).
- Using the equation from step 3, substitute the values v_final = 0, m_bullet = 0.025 kg, and m_pendulum = 2.25 kg to find v_bullet.
C. The percentage of energy lost:
- To calculate the percentage of energy lost, we need to compare the initial kinetic energy of the bullet with the final kinetic energy of the bullet and pendulum system.
- The initial kinetic energy of the bullet is given by: KE_initial = (1/2) * m_bullet * v_bullet^2, where m_bullet is the mass of the bullet determined in step B, and v_bullet is the initial velocity of the bullet determined in step B.
- The final kinetic energy of the system (bullet + pendulum) is given by: KE_final = (1/2) * (m_bullet + m_pendulum) * v_final^2, where m_bullet and m_pendulum are the masses determined in step B, and v_final is the final velocity of the system determined in step A.
- Calculate the values of KE_initial and KE_final using the given values.
- The energy lost is given by: Energy_lost = KE_initial - KE_final.
- Finally, calculate the percentage of energy lost by dividing Energy_lost by KE_initial and multiplying by 100.
By following these steps, you will be able to find the solution to this problem.