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 can use the principle of conservation of linear momentum and conservation of mechanical energy.
A. Speed of the system after collision:
1. We need to find the final velocity of the system after the collision. We can use the conservation of momentum equation:
(m1 * v1) + (m2 * v2) = (m1 + m2) * vf, where m1 is the mass of the bullet, m2 is the mass of the pendulum, v1 is the initial velocity of the bullet, v2 is the initial velocity of the pendulum, and vf is the final velocity of the system.
We know m1 = 25 g = 0.025 kg and m2 = 2.25 kg.
Assuming the pendulum is initially at rest (v2 = 0), the equation simplifies to:
(0.025 kg * v1) + (2.25 kg * 0) = (0.025 kg + 2.25 kg) * vf.
Simplifying further:
0.025 kg * v1 = 2.275 kg * vf.
2. We also know that the momentum before the collision is zero since the bullet is fired into the pendulum at rest.
Therefore, v1 = 0.
The equation becomes:
0.025 kg * 0 = 2.275 kg * vf.
0 = 2.275 kg * vf.
vf = 0 m/s.
Hence, the speed of the system after the collision is 0 m/s.
B. Initial speed of the bullet:
1. We can use the conservation of mechanical energy to find the initial speed of the bullet.
The formula for the potential energy of the pendulum is given by:
Potential Energy (PE) = mass * gravity * height.
The bullet embedded in the pendulum caused the pendulum to rise to a height of 10 cm = 0.1 m.
The gravitational acceleration (g) is approximately 9.8 m/s².
The potential energy of the pendulum is:
PE = 2.25 kg * 9.8 m/s² * 0.1 m.
2. The bullet transfers its kinetic energy to potential energy when it becomes embedded in the pendulum.
The initial kinetic energy of the bullet is equal to the transferred potential energy:
KE_bullet = PE = 2.25 kg * 9.8 m/s² * 0.1 m.
3. The formula for kinetic energy is given by:
Kinetic Energy (KE) = 0.5 * mass * velocity².
We need to find the initial velocity of the bullet (v1):
0.5 * 0.025 kg * v1² = 2.25 kg * 9.8 m/s² * 0.1 m.
Simplifying further:
0.0125 * v1² = 2.205.
v1² = 2.205 / 0.0125.
v1² = 176.4.
v1 = √(176.4) ≈ 13.27 m/s.
Therefore, the initial speed of the bullet is approximately 13.27 m/s.
C. Percentage of energy lost in the process:
1. The initial kinetic energy of the system (bullet and pendulum) is given by the sum of their individual kinetic energies:
KE_initial = 0.5 * (0.025 kg * (13.27 m/s)² + 2.25 kg * 0 m/s) = 0.5 * (0.5637 + 0) = 0.28185 J.
2. The final kinetic energy of the system is zero since the system comes to rest after the collision.
KE_final = 0 J.
3. The energy lost during the process is the difference between the initial and final kinetic energies:
Energy lost = KE_initial - KE_final = 0.28185 J - 0 J = 0.28185 J.
4. To find the percentage of energy lost, we can use the following formula:
Energy lost percentage = (Energy lost / KE_initial) * 100.
Energy lost percentage = (0.28185 J / 0.28185 J) * 100 = 100%.
Therefore, the percentage of energy lost in the process is 100%.