Stepwise Derivation of the Law of Conservation of Momentum with Numericals
Linear momentum is a fundamental quantity in physics, defined as the product of mass and velocity of an object. If a system is made up of several bodies or particles, each with its own mass and velocity, the total linear momentum of the system is the vector sum of individual momenta. This concept forms the base of various topics in mechanics and is crucial for understanding motion, collisions, and force interactions.
What is Conservation of Linear Momentum?
The law of conservation of linear momentum states that the total linear momentum of an isolated system remains constant if no external force acts on the system. In any physical scenario where the system does not interact with external influences, the sum of all the individual momenta before an event (like a collision) equals the sum after the event.
This law is directly derived from Newton’s Second and Third Laws of Motion. It is a cornerstone for analyzing all types of collisions and processes involving changes in velocity within a system.
Derivation of the Law of Conservation of Momentum
For a system of n particles with masses m1, m2, m3, ..., mn and velocities v1, v2, v3, ..., vn, the total linear momentum is:
According to Newton’s Second Law of Motion, the rate of change of momentum equals the net external force:
For an isolated system, F = 0. Thus,
This means the total linear momentum remains unchanged if no external force is applied.
Detailed Stepwise Derivation using Two-Body Collision
Consider two bodies, A and B, with masses m1 and m2. They move initially with velocities u1 and u2 and after a collision, their velocities become v1 and v2 respectively.
- Let FAB be the force exerted by B on A.
- Let FBA be the force exerted by A on B.
According to Newton’s Third Law,
Impulse is force multiplied by time of interaction (Δt). So,
Impulse on B: FBA ⋅ Δt = m2v2 – m2u2
Since FAB = – FBA,
Or,
Thus, the total linear momentum before collision equals the total after collision for an isolated system. Visit Law of Conservation of Momentum Derivation for more details.
Key Formulas and Table
| Quantity | Formula | SI Unit |
|---|---|---|
| Linear Momentum (p) | p = m × v | kg·m/s |
| Conservation of Momentum | m1u1 + m2u2 = m1v1 + m2v2 | kg·m/s |
| Impulse | Impulse = F × Δt = change in momentum | N·s = kg·m/s |
| Force and Momentum Relation | F = dp/dt | N (Newton) |
Applications of Conservation of Linear Momentum
- Collisions: Both elastic and inelastic collisions conserve momentum within closed systems. Learn about collision types.
- Rocket propulsion: Rocket expels gases and moves in the opposite direction.
- Recoil of a gun: When a bullet is fired, the gun moves backward to maintain total momentum.
- Motion of asteroids or space debris after impact in the absence of external forces.
Example Problems and Stepwise Solutions
Example 1: A 2 kg object moving at 3 m/s collides with a 3 kg object at rest. If after collision they join together, what is their common velocity?
- Initial momentum: (2×3) + (3×0) = 6 kg·m/s
- Let final velocity be v, total mass = 2+3 = 5 kg
- Using conservation: 6 = 5×v ⇒ v = 1.2 m/s
- The combined object moves at 1.2 m/s.
Example 2: Two objects of masses 4 kg and 2 kg move towards each other with velocities 5 m/s and -3 m/s respectively. What is their total momentum before collision?
- Total momentum = (4 × 5) + (2 × -3) = 20 - 6 = 14 kg·m/s
For more application-based problems, check the Conservation of Momentum page.
Stepwise Approach to Solving Conservation of Momentum Problems
| Step | Description |
|---|---|
| 1 | Identify all masses and their velocities before the event. |
| 2 | Write the expression for total initial momentum. |
| 3 | Write the expression for total final momentum. |
| 4 | Equate the two momenta (if no external force acts). |
| 5 | Solve for the unknown quantity. |
Summary Notes
- If no external force acts on a system, total linear momentum remains conserved.
- Momentums are vectors; direction must be carefully assigned.
- Conservation applies before and after collisions in a straight line.
- Study related topics: Impulse-Momentum Theorem.
Practice Questions
- Two bodies of masses 2 kg and 5 kg moving in a straight line strike each other. Describe what happens to their total momentum.
- A gun of mass 6 kg fires a bullet of mass 0.06 kg with a velocity of 100 m/s. Find the recoil velocity of the gun.
- Explain why the law of conservation of momentum does not hold in the presence of friction.
Next Steps & Deep Dive
- Explore advanced applications at Conservation of Linear Momentum.
Mastering the law of conservation of momentum gives strong problem-solving skills for exams and established understanding for all advanced mechanics concepts.
FAQs on Law of Conservation of Momentum: Derivation, Formula & Examples
1. What is the law of conservation of momentum with derivation?
The law of conservation of momentum states that if no external force acts on a system of interacting bodies, the total momentum of the system remains constant.
Derivation steps:
• Consider two bodies with masses m1 and m2, initial velocities u1 and u2.
• After collision, velocities become v1 and v2.
• Using Newton’s Third Law: force on the first equals minus the force on the second.
• Impulse equals change in momentum: m1v1 – m1u1 = – (m2v2 – m2u2)
• Rearranging: m1u1 + m2u2 = m1v1 + m2v2
The total linear momentum before collision equals the total after collision.
2. Where does the law of conservation of momentum come from?
The law of conservation of momentum is derived from Newton’s Second and Third Laws of Motion:
• Newton’s Second Law relates force to rate of change of momentum.
• Newton’s Third Law states for every action, there is an equal and opposite reaction.
• Together, these mean that in an isolated system (no external force), the momentum lost by one object is gained by the other, keeping total momentum constant.
3. State law of conservation of momentum and derive its mathematical expression.
Statement: If no external force acts on a system, the total momentum remains conserved.
Derivation:
• For two bodies:
Initial momentum = m1u1 + m2u2
Final momentum = m1v1 + m2v2
Setting them equal: m1u1 + m2u2 = m1v1 + m2v2
This is the law’s mathematical form.
4. What is the law of conservation of momentum formula?
The formula for the law of conservation of momentum (for two bodies):
m1u1 + m2u2 = m1v1 + m2v2
Where:
• m = mass of body
• u = initial velocity
• v = final velocity
This means total momentum before interaction equals total after interaction (if no external force).
5. How do you prove law of conservation of momentum?
Proof involves applying Newton’s laws:
1. Take two bodies A and B, with no external force.
2. During collision, force on A due to B equals minus force on B due to A.
3. Calculate impulse (change in momentum) for each body.
4. Add the two equations; external impulses cancel.
5. The result shows total momentum does not change.
This demonstrates momentum is conserved in an isolated system.
6. What are examples of conservation of momentum in daily life?
Examples of conservation of momentum include:
• Recoil of a gun: After firing, gun moves backward, bullet moves forward, total momentum is conserved.
• Jumping from a boat: The boat moves in the opposite direction.
• Ice skaters pushing off each other: Skaters move in opposite directions with equal and opposite momenta.
• Two billiard balls colliding: Their total momentum remains constant before and after collision.
7. Does conservation of momentum apply in all types of collisions?
Yes, conservation of momentum holds for all collisions (elastic, inelastic, and perfectly inelastic), as long as there are no external forces acting on the system.
• In elastic collisions: both momentum and kinetic energy are conserved.
• In inelastic collisions: only momentum is conserved; kinetic energy is not.
8. What is the difference between elastic and inelastic collisions with respect to momentum?
In both elastic and inelastic collisions, momentum is conserved.
• Elastic collision: Both momentum and kinetic energy are conserved.
• Inelastic collision: Only momentum is conserved; kinetic energy is partly lost as heat, sound, or deformation.
Momentum’s conservation is a universal principle in collisions, regardless of their type.
9. What are the main conditions for the law of conservation of momentum to hold?
The law holds true under the following conditions:
• The system is isolated (no external force acts).
• Interaction occurs only between bodies within the system.
• Masses remain constant during the event.
If these conditions are met, linear momentum is conserved.
10. What is the importance of conservation of momentum in Physics?
Conservation of momentum is a fundamental principle that explains:
• The behavior of objects during collisions and explosions.
• Predicts outcomes in problems involving moving bodies.
• Forms the basis for understanding rocket motion, bullet recoil, and particle interactions.
• Is crucial for solving physics problems at both board and competitive exam levels.
11. What is impulse and how is it related to momentum?
Impulse is the product of force and the time for which it acts (Impulse = F × Δt).
• It equals the change in momentum of a body.
• Units of impulse are the same as momentum (kg·m/s or N·s).
• Formula: Impulse = Change in Momentum = m(v – u)
12. Is angular momentum also conserved like linear momentum?
Yes, angular momentum is conserved if no external torque acts on a system.
• Angular momentum (L) formula: L = r × p.
• Applications: Figure skaters pulling arms in to spin faster, planetary motion.
• Like linear momentum, angular momentum remains constant unless acted upon by an external torque.
