Momentum Calculator

Momentum is the product of an object's mass and velocity. It's a vector quantity (has direction) and is conserved in closed systems. **The Formula**: p = m × v Where: - p = momentum (kg·m/s) - m = mass (kg) - v = velocity (m/s, vector) Since velocity is a vector, momentum is also a vector. It has the same direction as velocity. **Units**: - **SI**: kg·m/s or N·s (equivalent) - **Imperial**: slug·ft/s or lb·s **Conservation of Momentum**: In a closed system (no external forces), total momentum is conserved: **p_total (before) = p_total (after)** This is one of the most important laws in physics. It applies to: - Collisions (cars, billiards, particles) - Explosions (recoil) - Rocket propulsion - Particle physics **Types of Collisions**: **1. Elastic Collision**: - Both momentum AND kinetic energy conserved - Objects bounce apart perfectly - Examples: billiard balls (approx), atomic collisions **2. Inelastic Collision**: - Momentum conserved, KE not fully conserved - Some KE becomes heat, sound, deformation - Examples: most real collisions **3. Perfectly Inelastic Collision**: - Objects stick together after collision - Maximum KE loss while conserving momentum - Examples: arrow sticking in target, cars crumpling together **Momentum vs Energy**: | Quality | Momentum | Kinetic Energy | |---------|----------|----------------| | Formula | p = mv | KE = ½mv² | | Vector? | Yes | No (scalar) | | Always conserved? | Yes (closed systems) | Not always | | Depends on velocity | Linearly | Quadratically | Momentum scales linearly with velocity; kinetic energy scales with v². **Impulse**: Impulse is change in momentum: Impulse = Δp = F × Δt This means force applied over time changes momentum. Safety devices work by increasing Δt to reduce force: - **Airbags**: Extend time to stop, reducing force - **Crumple zones**: Deform to increase stopping distance - **Catcher's mitt**: Moves back to catch ball gently - **Rubber floor mats**: Absorb impact energy **Real-World Examples**: **Example 1**: Car momentum 1500 kg car at 20 m/s (72 km/h) p = 1500 × 20 = 30,000 kg·m/s **Example 2**: Bullet 10 g bullet at 400 m/s p = 0.01 × 400 = 4 kg·m/s **Example 3**: Person walking 70 kg person at 1.5 m/s p = 70 × 1.5 = 105 kg·m/s The car has much more momentum than the bullet (30,000 vs 4) despite lower velocity, because of mass. **Conservation Example: Collision** Car A (1500 kg, 20 m/s) hits stationary Car B (1000 kg): Before: p = 1500 × 20 + 1000 × 0 = 30,000 kg·m/s If they stick together: Total mass = 2500 kg After: p = 2500 × v = 30,000 v = 12 m/s (velocity of combined wreckage) **Newton's Second Law in Momentum Form**: F = ma = m(dv/dt) = dp/dt This is actually the more general form of Newton's Second Law. Force equals the rate of change of momentum. For constant mass: F = ma For changing mass (rockets): F = (dm/dt)v + m(dv/dt) **Rocket Equation**: Rockets work by conservation of momentum: - Exhaust gas expelled backward - Rocket pushed forward (same momentum, opposite direction) - No 'push against anything' needed Tsiolkovsky rocket equation: Δv = v_exhaust × ln(m_initial / m_final) **Angular Momentum**: Rotational analog of linear momentum: L = I × ω (for rigid bodies) Where I = moment of inertia, ω = angular velocity. Angular momentum is also conserved — this is why: - Ice skaters spin faster when pulling arms in - Earth's rotation is stable - Helicopters need tail rotors **Relativistic Momentum**: At high speeds (significant fraction of c), use: p = γmv Where γ = 1/√(1-v²/c²) (Lorentz factor) For v << c, γ ≈ 1 and p ≈ mv (classical). **Photon Momentum**: Even massless photons have momentum: p = E/c = hf/c This is why light can push solar sails and why laser cooling works. **Common Scenarios**: **Billiards**: Nearly elastic collisions, momentum and energy transferred. **Car Crashes**: Inelastic collisions, momentum conserved, energy becomes deformation. **Rocket Launch**: Action-reaction, rocket gains forward momentum equal to exhaust backward momentum. **Recoil**: Gun and bullet have equal and opposite momenta (gun lighter, moves slower). **Ice Skating**: Skaters pushing apart gain equal and opposite momenta. **Tips for Problems**: 1. **Choose positive direction** consistently 2. **Include signs** for velocities (direction matters) 3. **Sum all momenta** in the system 4. **Use conservation** as primary tool 5. **Check units**: kg·m/s everywhere **Momentum Transfer**: In any interaction between two objects: - The momentum one loses, the other gains - This is Newton's Third Law (action-reaction) - Total system momentum is unchanged