Conservation of linear momentum

Law of conservation of linear momentum states that if there is no external force acting on two colliding objects, the momentum of the objects before the collision is equal to the momentum of the objects after the collision.

p_{1} + p_{2 }= p_{1} ’ + p_{2} ’ ………………….. Equation 1.4

m_{1 }v_{1} + m_{2} v_{2} = m_{1} v_{1 }’ + m_{2} v_{2 }’

If after collision both objects stick together,

m_{1 }v_{1} + m_{2} v_{2} = (m_{1 }+ m_{2} ) v’

where: m_{1} = mass of object 1, m_{2 }= mass of object 2, v_{1} = speed of object 1 before collision, v_{2} = speed of object 2 before collision, v_{1} ’ = speed of object 1 after collision, v_{2 }’ = speed of object 2 after collision, v’ = speed of both objects after collision

v or v’ has a positive sign if the object moves to the right and has a negative sign if the object moves to the left. If the objects’ motion directions are unknown, but both objects move in opposite directions, either of v or v’ has a positive sign while the other has a negative sign. If both objects move in the same direction, v or v’ of both objects has the same direction. The positive or negative sign indicates the direction of the object’s motion.

20 conceptual questions and answers about the conservation of linear momentum:

**1. Question:** What is linear momentum? **Answer:** Linear momentum, often just called momentum, is the product of an object’s mass and its velocity. It is a vector quantity and is represented by $p=mxv$.

**2. Question:** How does the conservation of linear momentum apply in collisions? **Answer:** The total momentum of a system before a collision is equal to the total momentum after the collision, provided no external forces act on the system.

**3. Question:** In what scenarios is linear momentum conserved? **Answer:** Linear momentum is conserved in all scenarios where the net external force on the system is zero.

**4. Question:** How does a rocket in space move forward given there’s no external force acting on it? **Answer:** A rocket in space expels gases backward, and due to the conservation of momentum, the rocket moves forward. The momentum of the expelled gases and the rocket remain equal and opposite, maintaining a net momentum of zero.

**5. Question:** Why does an ice skater spin faster when they pull their arms in? **Answer:** When the skater pulls their arms in, they reduce their moment of inertia. To conserve angular momentum, their rotational speed must increase.

**6. Question:** Can momentum be conserved if there is an external force acting on a system? **Answer:** For the duration the external force acts, momentum won’t be conserved. However, over short time intervals, the change might be negligible and momentum approximately conserved.

**7. Question:** What is an inelastic collision with respect to momentum? **Answer:** In an inelastic collision, momentum is conserved, but kinetic energy is not. Objects may stick together or deform during the collision.

**8. Question:** How can a system gain momentum without an external force? **Answer:** A system can’t gain momentum without an external force. Newton’s third law ensures that actions within a system have equal and opposite reactions, so net momentum remains unchanged.

**9. Question:** What is impulse and its relation to momentum? **Answer:** Impulse is the change in momentum of an object. It’s given by the product of the force acting on an object and the time duration over which it acts. Impulse is also a vector quantity.

**10. Question:** How do airbags in cars use the concept of momentum? **Answer:** Airbags increase the time over which a collision occurs. This reduces the force experienced by occupants, changing the momentum more gradually and reducing injury.

**11. Question:** How does bouncing differ from sticking in terms of momentum conservation in collisions? **Answer:** In both cases, momentum is conserved. However, when objects bounce, they may exchange more kinetic energy than when they stick together.

**12. Question:** Can a single object have conserved momentum? **Answer:** An isolated object’s momentum is conserved unless acted upon by an external force.

**13. Question:** How is momentum related to Newton’s third law? **Answer:** Newton’s third law states every action has an equal and opposite reaction. This ensures that momentum changes are balanced between interacting objects, leading to momentum conservation.

**14. Question:** Why does a bullet fired from a gun make the gun recoil? **Answer:** As the bullet gains forward momentum, the gun gains an equal amount of backward momentum due to the conservation of momentum.

**15. Question:** Can momentum be zero in a system with moving objects? **Answer:** Yes, if the vector sum of all individual momenta is zero. For instance, two objects of equal mass and speed moving in opposite directions have zero net momentum.

**16. Question:** How does the center of mass motion relate to momentum conservation? **Answer:** The motion of a system’s center of mass is directly tied to the system’s total momentum. If no external forces act, the center of mass will move at a constant velocity.

**17. Question:** Is momentum always conserved in explosions? **Answer:** Yes, the total momentum before an explosion (usually zero) is equal to the total momentum after, even though the pieces may be moving in different directions.

**18. Question:** How do pool players use momentum conservation principles? **Answer:** When striking pool balls, players anticipate how balls will move post-collision based on momentum conservation, allowing them to predict shot outcomes.

**19. Question:** Why doesn’t a car come to a stop immediately after turning off the engine? **Answer:** Due to its momentum. Unless acted upon by an external force (like friction), it will continue moving.

**20. Question:** In a two-object system, if one object’s momentum increases, what happens to the other’s momentum? **Answer:** The other object’s momentum will decrease by an equal amount in the opposite direction, keeping the total momentum conserved.

Understanding the conservation of linear momentum is foundational in physics and has applications ranging from basic mechanics to advanced fields of study.