Physics: Momentum and Collisions

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