Conservation of Linear Momentum in Collisions: Key Applications & Examples
The Law of Conservation of Linear Momentum is a core principle in physics, especially in the study of mechanics. It explains how, in the absence of external forces, the total momentum of a system remains unchanged, even when objects within the system interact, collide, or exchange forces. This concept is essential for understanding a wide variety of real-world physical phenomena, from simple collisions on a billiard table to the propulsion of rockets.
Definition and Principle of Conservation of Linear Momentum
Linear momentum is defined as the product of an object's mass and its velocity (momentum p = m × v). The law of conservation of linear momentum states that:
- If no external force acts on a system of objects, the total linear momentum of the system remains constant.
This means that in a closed or isolated system, momentum cannot be created or destroyed—only transferred among objects in the system.
Newton’s Third Law and Momentum Conservation
Newton’s third law forms the basis for understanding momentum conservation. It states: Whenever one object exerts a force on another, the second object exerts an equal and opposite force back. Mathematically:
If both forces act over the same time interval, the impulse (force × time) felt by each object is equal and opposite. Since impulse equals the change in momentum, both objects experience equal but opposite changes in momentum.
mA × ΔvA = –mB × ΔvB
Derivation of Conservation of Linear Momentum
Consider two objects, A and B, with masses mA and mB, and initial velocities uA and uB, respectively. After interaction (such as a collision), their velocities become vA and vB.
Change in momentum of B: mB(vB – uB)
According to Newton’s third law, the total change in the system’s momentum is zero:
This simplifies to:
So, the sum of the objects' momenta before the interaction equals the sum after, confirming the conservation of momentum.
Collisions and Applications
The law of conservation of momentum is crucial in analyzing collisions—both elastic and inelastic. In all cases, the total momentum of the system is conserved if no external force intervenes.
- Same mass, same velocity collision: Both objects retain their velocity; total momentum remains unchanged.
- Different masses, same velocity collision: The heavier object’s velocity decreases, transferring momentum to the lighter one, but overall momentum is constant.
- Collision with one object at rest: Momentum redistributes so total system momentum is the same before and after.
Real-World Examples
- Gun Recoil: When a bullet is fired, the gun recoils backward, conserving total momentum.
- Rocket Propulsion: Expulsion of gas downwards makes the rocket move upwards, due to momentum conservation.
- Colliding Balls/Ice Hockey: When two balls (or hockey pucks) collide, regardless of breaking or changing direction, their combined momentum stays constant.
Core Formulas Table
| Concept | Formula | Description |
|---|---|---|
| Linear Momentum (p) | p = m × v | Product of mass (kg) and velocity (m/s) |
| Momentum Conservation (two objects) | mA uA + mB uB = mA vA + mB vB | Total momentum before = total after |
| Impulse | Impulse = Δp = F × Δt | Change in momentum equals force times time interval |
Step-by-Step Problem Solving Approach
| Step | Action | Purpose |
|---|---|---|
| 1 | List masses and velocities of all objects before and after the event. | Establish variables for calculation. |
| 2 | Apply the momentum conservation formula. | Set up the mathematical equation. |
| 3 | Substitute known values and solve for the unknown. | Arrive at the solution in a systematic way. |
| 4 | Check if the final answer is physically meaningful (direction, magnitude). | Avoid common sign and interpretation mistakes. |
Sample Numericals
Example 1: Two objects each of mass 10 kg move towards each other at 5 m/s. After a head-on elastic collision in an isolated system, what are their velocities?
By conservation, total final momentum must also be 0. So, if they collide head-on elastically with the same mass and velocity, velocities remain unchanged: 5 m/s (opposite direction).
Example 2: Find the recoil velocity of a 20 kg gun after firing a 0.1 kg bullet at 200 m/s.
(20 × v) = 0.1 × 200 → v = (0.1 × 200)/20 = 1 m/s (opposite to bullet motion).
When is Kinetic Energy Not Conserved but Momentum Is?
In inelastic collisions, such as when two objects stick together or deform, kinetic energy is lost as heat or sound. However, the total linear momentum of the system remains conserved.
Related Vedantu Resources and Further Learning
- Conservation of Linear Momentum – Full Topic Guide
- Impulse-Momentum Theorem Explained
- Elastic & Inelastic Collisions
Practice Questions
- Two bodies, each of 5 kg, move towards each other at 2 m/s. Calculate their velocities post-collision, assuming a perfectly elastic collision.
- A person jumps out of a boat. How does the boat move and why?
- Describe a scenario where momentum is conserved but kinetic energy is not.
What’s Next?
- Revise the momentum formulas regularly.
- Practice more problems from Conservation of Momentum and Law of Conservation of Momentum Derivation.
- Explore applications in collisions in one and two dimensions.
For further study, refer to advanced concepts like work, energy, and power, and continue practicing real-world momentum problems to solidify your understanding.
FAQs on Conservation of Linear Momentum: Concept, Laws & Formulas
1. What is the Law of Conservation of Linear Momentum?
The Law of Conservation of Linear Momentum states that the total linear momentum of a system remains constant if no external force acts on it. This means:
- The sum of momenta before an event (such as a collision) equals the sum after the event
- Formula: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
- Applicable for all isolated systems, as per the current CBSE and NTA syllabus
2. How do you prove the law of conservation of momentum mathematically?
To prove the law mathematically:
- Start with Newton’s Third Law: FAB = -FBA
- The time duration of action and reaction is the same
- Impulse (change in momentum) on both objects is equal and opposite
- m₁(u₁ - v₁) = -m₂(u₂ - v₂)
- On rearranging: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
3. Is momentum always conserved in all types of collisions?
Momentum is always conserved in all types of collisions, provided there is no external force acting on the system.
- Elastic collisions: Both momentum and kinetic energy are conserved
- Inelastic collisions: Momentum is conserved, but kinetic energy is not
4. What is linear momentum? Give its formula.
Linear momentum (p) is a physical quantity that is the product of an object’s mass and velocity.
Formula: p = m × v
Where m = mass (kg) and v = velocity (m/s). The SI unit is kg·m/s.
5. What are some real-life applications of the conservation of linear momentum?
Key applications of conservation of momentum include:
- Recoil of a gun: When a gun is fired, the gun moves backward as the bullet moves forward.
- Rocket propulsion: Rockets move upward by expelling gases downward.
- Collisions in sports: For example, when two ice skaters push away from each other, they move in opposite directions.
6. Where does the conservation of linear momentum not apply?
The law does not apply if there is any external force acting on the system.
- Examples include cases with significant friction, air resistance, or externally applied forces
- Only in isolated systems (no external force), momentum is conserved
7. How is the impulse-momentum theorem related to conservation of momentum?
The impulse-momentum theorem states that the change in momentum of an object equals the impulse applied to it.
- Impulse = Change in momentum = F × Δt
- When total external force is zero, total momentum does not change—leading to the law of conservation of momentum
8. What happens when two objects of different masses collide?
When two objects with different masses collide:
- Momentum is redistributed based on mass and velocity
- The heavier object's velocity usually changes less than the lighter object
- Total momentum before = total momentum after the collision
9. Can kinetic energy be lost during momentum-conserving events?
Yes, kinetic energy can be lost in some collisions—even when momentum is conserved.
- Inelastic collisions: kinetic energy is not conserved but momentum is
- Energy may transform into heat, sound, or deformation
10. How do you solve numerical problems using conservation of momentum?
Follow these steps:
- List all masses, initial, and final velocities
- Write the momentum conservation equation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
- Substitute known values
- Solve for the unknown variable (usually velocity or mass)
11. What is an example of momentum being conserved but kinetic energy not being conserved?
An inelastic collision is a classic example:
- When two cars crash and stick together, total momentum is conserved
- Kinetic energy decreases due to deformation, heat, or sound
12. Why is the law of conservation of momentum important for JEE and NEET exams?
This law is a foundation for solving physics questions in competitive exams.
- Frequently asked in questions involving collisions, explosions, and system dynamics
- Directly applies to both theoretical concepts and numerical problems in JEE and NEET syllabi
