Chapter 8: Problem 43
A 12.0-g rifle bullet is fired with a speed of 380 m/s into a ballistic pendulum with mass 6.00 kg, suspended from a cord 70.0 cm long (see Example 8.8 in Section 8.3). Compute (a) the vertical height through which the pendulum rises, (b) the initial kinetic energy of the bullet, and (c) the kinetic energy of the bullet and pendulum immediately after the bullet becomes embedded in the wood.
Short Answer
Step by step solution
Understand the Conservation Law Applicable
Apply Conservation of Momentum for the Collision
Calculate the Vertical Height Using Conservation of Energy
Compute the Initial Kinetic Energy of the Bullet
Find the Kinetic Energy after the Collision
Conclusion
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conservation of Momentum
When the bullet is fired into the pendulum, it carries momentum, while the pendulum is initially at rest and has no momentum. The equation for conservation of momentum can be set up as follows:
- Before collision: Bullet momentum = \( m_b \cdot v_b \)
- After collision: System momentum = \( (m_b + M) \cdot v_f \)
Conservation of Energy
Immediately after the collision, the system's energy is purely kinetic. As it swings upwards, this kinetic energy is transformed into gravitational potential energy. The conservation of energy equation can be laid out as:
- Initial kinetic energy: \( \frac{1}{2} (m_b + M) v_f^2 \)
- Potential energy at the highest point: \( (m_b + M)gh \)
Kinetic Energy
Initially, the bullet has kinetic energy given by \( KE = \frac{1}{2} m_b v_b^2 \). This energy allows the bullet to penetrate the pendulum. Once the bullet is embedded, the combined system's kinetic energy is calculated based on their final velocity \( v_f \). The drastic difference in kinetic energy before and after the collision reflects the energy loss mainly due to deformation and heat, demonstrating the inelastic nature of the collision.
Potential Energy
As the pendulum rises to its maximum height, its velocity momentarily drops to zero, and all the mechanical energy is stored as potential energy. This allows us to calculate how high the pendulum reaches by equating it with the kinetic energy just after the collision. Therefore, potential energy gives a clear measure of the system's energy state when motion halts during the swing, highlighting the energy transfer between types and the conservation throughout the system's motion.