Physics: Momentum and Collisions

Study Notes

Calculations Involving Velocity

The provided notes use different algebraic manipulations to reach the equation $u_1 + v_1 = \sqrt{u_2^2 + v_2^2}$
This is likely the final equation used in the physics problem

Isolated System

Momentum is conserved in an isolated system
The total momentum of the system before collision (Pt) equals the total momentum of the system after the collision (Pi)
This means that the momentum of the system remains constant over time
The equation for this is: m1u1 + m2u2 = m1v1 + m2v2
Where m1 and m2 represent the masses of two objects colliding, u1 and u2 are their initial velocities, and v1 and v2 are their final velocities
This is the basic momentum conservation equation. It states that the momentum of the system before the collision (m1u1 + m2u2) must equal the momentum of the system after the collision (m1v1 + m2v2)

Elastic Collision

The collision type where the total kinetic energy (KE) before the collision equals the total KE after the collision
There is no loss of KE in the form of heat, sound, or any other forms of energy
Ideal collisions of molecules in gases can be considered elastic since they involve very small amounts of energy loss

Unit Conversion

1kgms⁻¹ = 1N⋅s

Law of Conservation of Momentum

The total momentum before a collision is equal to the total momentum after the collision
This means that the momentum of a system remains constant over time, regardless of how the objects within the system interact

Isolated System

A system is isolated if no external forces act on it
An isolated system has no external forces acting on it, resulting in a constant momentum

Proof of the Law of Conservation of Momentum

The law states that the change in momentum of a system is equal to the impulse applied to that system.
The impulse is defined as the product of the force acting on the system and the time interval over which it acts
$F_{ext} = \frac{\Delta P}{\Delta t}$, where Fext is the external force, ΔP is the change in momentum, and Δt is the time interval.
For an isolated system, Fext = 0, and therefore, ΔP = 0. This means that the momentum of the system is conserved, as the change in momentum is zero.

Example: Gun and Bullet

The initial momentum of the system (gun and bullet) is zero as they are at rest.
$P_i = 0$
When the bullet is fired, it gains momentum
For the momentum of the system to remain constant, the gun recoils backwards, generating an equal and opposite momentum to that of the bullet.
This ensures that the total momentum of the system remains zero.

Example: Rocket Propulsion

The rocket engines burn fuel and expel hot gas downwards, creating a force that pushes the rocket upwards.
This force generates an equal and opposite reaction force that propels the rocket forward
The momentum of the expelled gas and the rocket moving forward is equal and opposite, thus conserving the momentum of the system.

Elastic Collision in One Dimension

This type of collision occurs when two objects collide in a straight line, and their total kinetic energy is conserved.
This means that there is no energy loss due to heat or sound
The KE before the collision is the same as the KE after the collision

Law of Conservation of Momentum

The total momentum of a system remains constant over time, regardless of the interactions between its components.
Before collision, momentum is m1v1 + m2v2
After collision, momentum is still m1v1 + m2v2

Momentum

The measure of an object's mass in motion
Calculated by multiplying the object's mass by its velocity
Unit of momentum: kgms⁻¹

Description

This quiz covers calculations involving velocity, momentum conservation in isolated systems, and the principles of elastic collisions. Understand the equations governing these concepts and test your knowledge with various problems related to momentum. Prepare to apply algebraic manipulations to solve for velocities and masses during collisions.