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Two-body collisions - general concepts
No external forces. Collision via massless springs or other force type.
1.1 Momentum exchange and impulse
During collision: internal force causes change of momentum 
- Momentum conserved.
1.2 An off-axis collision in 2D
Elastic collisions in the Lab frame
Conservation of momentum: 
Conservation of energy:
[Note that the motion is in a plane, and the 2D representation can be trivially extended into 3D by rotation of the plane].
2.1 Elastic collisions in 1D in the Lab frame
→ Relative speed before collision = Relative speed after
2.2 Special case in 1D where target particle is at rest
Special cases:
Example: Newton’s cradle
- If the balls are touching, the most general case is:
Momentum after collision :
Energy after collision :
2 equations, 3 unknowns - The obvious solution:
- But other solution(s) possible:
Momentum : 
Energy :
So why does the simple solution always prevail?
Relationship between speeds and angles
Note we can also do this via components of momentum :
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FAQs on Two Body Collisions in Lab Frame - Collisions, Classical Mechanics, CSIR-NET Physical Sciences - Physics for IIT JAM, UGC - NET, CSIR NET
| 1. How do you define a two-body collision in the lab frame? |  |
| 2. What is the significance of studying two-body collisions in classical mechanics? | |
Ans. Studying two-body collisions in classical mechanics allows us to understand the principles of conservation of momentum and energy. It helps in predicting and analyzing the outcome of the collision, such as the final velocities and angles of the particles involved. This understanding is crucial in various fields like engineering, astrophysics, and particle physics.
| 3. How is the outcome of a two-body collision determined in the lab frame? | |
Ans. The outcome of a two-body collision in the lab frame is determined by applying the laws of conservation of momentum and energy. By considering the initial velocities, masses, and types of interaction between the particles, we can calculate the final velocities and angles of the particles after the collision.
| 4. Can you provide an example of a two-body collision in the lab frame? | |
Ans. Sure! Let's consider a scenario where two billiard balls collide on a frictionless table. If the first ball is initially at rest and the second ball is moving towards it with a certain velocity, the collision between them can be treated as a two-body collision in the lab frame. By analyzing the masses, initial velocities, and types of collision, we can determine the final velocities and directions of the balls after the collision.
| 5. How does the lab frame differ from the center of mass frame in two-body collisions? | |
Ans. The lab frame is the reference frame where the observer is stationary and measures the motion of the colliding particles. In contrast, the center of mass frame is a reference frame where the total momentum of the system is zero. In two-body collisions, the lab frame is used to analyze and predict the outcome of the collision, while the center of mass frame simplifies the calculations by eliminating the relative motion between the particles. Both frames provide valuable insights into the collision dynamics, but their perspectives and calculations differ.
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