Inelastic Collision to SHM: Why Energy Conservation Fails During Impact

Simulation Overview: This simulation visualizes the classic Ballistic Pendulum problem using a spring system. To solve this correctly, you must separate the event into two distinct physical phases. You cannot use a single energy equation to solve this problem because energy is lost during the collision.

The Physics Breakdown:

1. Phase 1: The Collision (Conservation of Momentum) During the perfectly inelastic collision, the bullet embeds into the block. Kinetic energy is NOT conserved here (lost to heat/deformation), but momentum is.

• Formula: m·v₀ = (M + m)·v_f
• Result: This gives you the Post-Impact Velocity (v_f).

1. Phase 2: The Oscillation (Conservation of Energy) Once the bullet is embedded, the system becomes a Simple Harmonic Oscillator. Now, mechanical energy is conserved. The kinetic energy of the combined mass converts entirely to elastic potential energy at the maximum amplitude.

• Formula: ½(M + m)(v_f)² = ½k(A)²
• Result: This allows you to solve for Maximum Amplitude (A).

Key Simulation Variables:

• Block Mass (M): The stationary mass attached to the spring.
• Bullet Mass (m): The projectile mass.
• Combined Mass (M + m): The total inertia used for the oscillation phase.
• Spring Constant (k): The stiffness of the spring (N/m).
• Impulse Approximation: We assume the collision is instantaneous (Δt → 0), meaning the spring exerts zero force during the actual impact.

Common Student Mistakes:

• The Energy Trap: Trying to use ½mv₀² = ½kA². This is incorrect because it ignores the energy lost when the bullet drills into the block.
• Mass Confusion: Forgetting to add the bullet's mass to the block (M + m) when calculating the period or energy of the spring system.

Frequently Asked Questions (FAQ):

Q: Why is Kinetic Energy not conserved during the collision? A: Because it is a perfectly inelastic collision. A significant amount of energy is converted into non-mechanical forms like heat and sound as the bullet deforms the block.

Q: Does the spring start compressing before the bullet stops? A: In reality, yes, slightly. However, in physics problems, we use the "Impulse Approximation," assuming the collision happens so fast that the block hasn't moved yet, so the spring force is effectively zero during the impact.

Q: How do I find the Period (T) of this system? A: Use the formula for a spring mass system, but ensure you use the total mass: T = 2π√((M+m)/k).