Linear Momentum Formulas
| Formula Name | Formula | Variables | Explanation |
|---|---|---|---|
| Basic Linear Momentum | p = mv | p = momentum (kg⋅m/s) m = mass (kg) v = velocity (m/s) |
The fundamental formula expressing momentum as the product of mass and velocity |
| Momentum from Force | p = F⋅t | p = momentum (kg⋅m/s) F = force (N) t = time (s) |
Momentum equals impulse (force × time) |
| Change in Momentum | Δp = m⋅Δv | Δp = change in momentum m = mass (kg) Δv = change in velocity (m/s) |
Change in momentum for constant mass |
| Variable Mass Momentum | p = Σ(mᵢvᵢ) | p = total momentum mᵢ = individual masses vᵢ = individual velocities |
Total momentum of a system with multiple objects |
| Relativistic Momentum | p = γmv | γ = Lorentz factor = 1/√(1-v²/c²) m = rest mass v = velocity c = speed of light |
Momentum at high velocities (near speed of light) |
Angular Momentum Formulas
| Formula Name | Formula | Variables | Explanation |
|---|---|---|---|
| Point Particle Angular Momentum | L = r × p = mvr sin θ | L = angular momentum (kg⋅m²/s) r = position vector (m) p = linear momentum θ = angle between r and v |
Angular momentum of a point mass |
| Rotational Angular Momentum | L = Iω | L = angular momentum (kg⋅m²/s) I = moment of inertia (kg⋅m²) ω = angular velocity (rad/s) |
Angular momentum for rotating rigid bodies |
| Orbital Angular Momentum | L = √[l(l+1)]ℏ | L = orbital angular momentum l = orbital quantum number ℏ = reduced Planck’s constant |
Quantum mechanical orbital angular momentum |
| Spin Angular Momentum | S = √[s(s+1)]ℏ | S = spin angular momentum s = spin quantum number ℏ = reduced Planck’s constant |
Intrinsic angular momentum of particles |
| Total Angular Momentum | J = L + S | J = total angular momentum L = orbital angular momentum S = spin angular momentum |
Combined orbital and spin angular momentum |
| Angular Momentum from Torque | L = τ⋅t | L = angular momentum τ = torque (N⋅m) t = time (s) |
Angular momentum equals angular impulse |
Conservation Laws
| Law | Formula | Conditions | Explanation |
|---|---|---|---|
| Conservation of Linear Momentum | p₁ + p₂ = p₁’ + p₂’ | No external forces | Total momentum before collision equals total momentum after |
| Conservation of Angular Momentum | L₁ + L₂ = L₁’ + L₂’ | No external torques | Total angular momentum remains constant |
| Elastic Collision (1D) | m₁v₁ + m₂v₂ = m₁v₁’ + m₂v₂’ | Perfectly elastic collision | Both momentum and kinetic energy conserved |
| Inelastic Collision | m₁v₁ + m₂v₂ = (m₁ + m₂)v’ | Perfectly inelastic collision | Objects stick together after collision |
Dimensional Formulas
| Quantity | Dimensional Formula | Base Units | Explanation |
|---|---|---|---|
| Linear Momentum | [M L T⁻¹] | kg⋅m⋅s⁻¹ | Mass × Length × Time⁻¹ |
| Angular Momentum | [M L² T⁻¹] | kg⋅m²⋅s⁻¹ | Mass × Length² × Time⁻¹ |
| Impulse | [M L T⁻¹] | N⋅s = kg⋅m⋅s⁻¹ | Same as momentum (Force × Time) |
| Angular Impulse | [M L² T⁻¹] | N⋅m⋅s = kg⋅m²⋅s⁻¹ | Same as angular momentum (Torque × Time) |
Related Concepts
| Concept | Formula | Variables | Relationship to Momentum |
|---|---|---|---|
| Impulse | J = F⋅Δt = Δp | J = impulse F = force Δt = time interval Δp = change in momentum |
Impulse equals change in momentum |
| Force from Momentum | F = dp/dt | F = force p = momentum t = time |
Newton’s second law in momentum form |
| Torque from Angular Momentum | τ = dL/dt | τ = torque L = angular momentum t = time |
Rotational analog of F = dp/dt |
| Kinetic Energy and Momentum | KE = p²/(2m) | KE = kinetic energy p = momentum m = mass |
Alternative expression for kinetic energy |
| Center of Mass Velocity | vₓₘ = (Σmᵢvᵢ)/Σmᵢ | vₓₘ = center of mass velocity mᵢ = individual masses vᵢ = individual velocities |
Related to total system momentum |
Relationships and Special Cases
Newton’s Laws and Momentum
- Newton’s First Law: In the absence of external forces, momentum remains constant
- Newton’s Second Law: F = ma = d(mv)/dt = dp/dt
- Newton’s Third Law: For every action, there is an equal and opposite reaction (momentum conservation)
Special Cases
- Zero Net External Force: Total momentum of system remains constant
- Collision Analysis: Use conservation of momentum to solve collision problems
- Rocket Propulsion: Momentum conservation explains rocket motion
- Rotational Systems: Angular momentum conserved when no external torques act
Units Summary
- SI Unit of Linear Momentum: kg⋅m/s (kilogram-meter per second)
- SI Unit of Angular Momentum: kg⋅m²/s (kilogram-meter squared per second)
- CGS Unit of Linear Momentum: g⋅cm/s (gram-centimeter per second)
- CGS Unit of Angular Momentum: g⋅cm²/s (gram-centimeter squared per second)
Study Tips for Students
- Remember the Vector Nature: Momentum is a vector quantity – direction matters
- Conservation Principles: Always check if external forces/torques are absent
- Sign Conventions: Establish consistent positive directions before solving
- Unit Analysis: Always verify units match expected dimensional formulas
- Problem-Solving Strategy: Identify the type of momentum problem first (linear vs angular, collision vs rotation)
This comprehensive guide covers all essential momentum formulas used in high school and introductory college physics courses, providing a solid foundation for understanding mechanical systems and collision dynamics.
Frequently Asked Questions (FAQs) about Momentum Fromulas
Q. What is the formula for momentum and how do you calculate it?
The basic formula for linear momentum is p = mv, where p represents momentum, m is the mass of the object (in kilograms), and v is its velocity (in meters per second). To calculate momentum, simply multiply the object’s mass by its velocity. For example, a car with mass 1000 kg moving at 20 m/s has momentum of 20,000 kg⋅m/s. Remember that momentum is a vector quantity, meaning it has both magnitude and direction.
Q. What is the SI unit of momentum and its dimensional formula?
The SI unit of momentum is kilogram-meter per second (kg⋅m/s) or equivalently Newton-second (N⋅s). The dimensional formula of linear momentum is [M L T⁻¹], which represents mass (M) multiplied by length (L) divided by time (T). For angular momentum, the SI unit is kg⋅m²/s with dimensional formula [M L² T⁻¹]. Understanding these units helps verify the correctness of calculations and ensures dimensional consistency in physics problems.
Q. What is the difference between linear momentum and angular momentum?
Linear momentum (p = mv) describes the motion of an object moving in a straight line or along a path, measuring the “quantity of motion” in translational movement. Angular momentum (L = Iω or L = mvr sin θ) describes rotational motion around an axis, measuring the “quantity of rotation.” The key differences are: (1) Linear momentum depends on mass and velocity, while angular momentum depends on moment of inertia and angular velocity, (2) Linear momentum is conserved when no external forces act, while angular momentum is conserved when no external torques act, and (3) Their dimensional formulas differ: [M L T⁻¹] for linear vs [M L² T⁻¹] for angular momentum.
Q. How do you calculate change in momentum (Δp)?
Change in momentum is calculated using the formula Δp = m(v_f – v_i) or Δp = mv_f – mv_i, where m is mass, v_f is final velocity, and v_i is initial velocity. This change in momentum equals the impulse applied to the object: Δp = F⋅Δt (force multiplied by time interval). For example, if a 2 kg ball changes velocity from 5 m/s to 15 m/s, the change in momentum is 2 × (15 – 5) = 20 kg⋅m/s. This concept is fundamental in analyzing collisions and understanding Newton’s second law.
Q. What is the law of conservation of momentum and when does it apply?
The law of conservation of momentum states that the total momentum of a closed system remains constant when no external forces act on it. Mathematically, for two objects: m₁v₁ + m₂v₂ = m₁v₁’ + m₂v₂’ (initial momentum equals final momentum). This law applies in all types of collisions—elastic and inelastic as long as no external forces interfere. It’s crucial for solving collision problems, understanding rocket propulsion, and explaining everyday phenomena like recoil when firing a gun. Even when objects stick together or kinetic energy is lost, momentum is always conserved in an isolated system.
Q. What is the formula for orbital and spin angular momentum in quantum mechanics?
In quantum mechanics, orbital angular momentum is given by L = √[l(l+1)]ℏ, where l is the orbital quantum number (l = 0, 1, 2, 3…) and ℏ (h-bar) is the reduced Planck’s constant (1.054 × 10⁻³⁴ J⋅s).
The spin angular momentum formula is S = √[s(s+1)]ℏ, where s is the spin quantum number (s = 1/2 for electrons). Unlike classical mechanics, quantum angular momentum is quantized and can only take specific discrete values. The z-component is given by L_z = m_l⋅ℏ (orbital) and S_z = m_s⋅ℏ (spin), where m_l and m_s are magnetic quantum numbers.