🚀 Impulse-Momentum Calculator & Simulator
- Specify Projectile Mass (M): Set the mass of the rigid body (0.5kg to 10kg) to drive the collision.
- Define Entry/Exit Collision Velocities: Adjust the entry velocity before obstacle impact (Vi) and the elastic rebound velocity (Vf) using the sliders.
- Fine-tune Cushioning Contact Time (dt): Set the contact time (sec) during which the colliding object is in contact with the obstacle. A longer contact time represents a cushioning effect that reduces the impact force.
- Run Real-time Collision Test: Click the [Start Collision Simulation] button to compare and analyze the peak curve variations of the impact force and the instantaneous structural deformation heat (glow) effect.
📚 Detailed Physics Explanation: Impulse and Momentum Integration Formulas ▼
1. The Physical Identity of the Impulse-Momentum Theorem
Integrating Newton's second law of motion (F = m·a) with respect to time t yields the Impulse-Momentum Theorem, which explains how force relates to changes in state of motion.
- Momentum (p): A physical quantity determined by the product of an object's mass and velocity, representing the magnitude of its motion.
p = m × v [kg·m/s] - Impulse (J): The total accumulation (integral) of force acting over a given period of time.
J = ∫ F dt = Δp [N·s]
This equation demonstrates that the impulse received by an object is perfectly equivalent to the change in its momentum.
2. Average Impact Force (F_avg) and the Crumple Zone Cushioning Formula
Even for collisions with the exact same change in momentum Δp, increasing the contact time Δt can drastically reduce the Average Collision Force acting on the object.
F_avg = J / Δt = m × (v_f - v_i) / Δt [N]
Vehicle crumple zones and compressed foam cushions in sports helmets are classic dynamic engineering technologies that force Δt to increase by 10 to 50 times as the structure deforms during impact, thereby reducing the average force F_avg transmitted to the human body to a safe range.
3. Calculation of Energy Dissipation Rate During Collision
The dissipation rate of kinetic energy released as plastic deformation and heat during a collision is measured as follows:
KE_loss = 1/2 × m × v_i^2 - 1/2 × m × v_f^2 [J]
In a perfectly inelastic collision where the exit velocity is 0, 100% of the incoming mechanical energy is absorbed (lost), causing maximum mechanical fatigue damage to the equipment. Therefore, while a higher rebound velocity reduces relative plastic structural damage, the impact experienced by the human body increases sharply due to the rapid acceleration in the opposite direction.