Impulse and Momentum

• Analyzing a situation from the perspective of momentum and impulse simplifies the forces to an average force, when the forces involved are not constant when two objects collide

Impulse & Momentum Defined

• Impulse is a vector described by the product of net force and time duration in a collision
• Impulse = FΔt
• Impulse Units: N⋅s
• Momentum is a vector product of an object's mass and velocity
• Momentum = p = mv
• Momentum Units: kg⋅m/s

Impulse-Momentum Theorem

• States that the impulse acting on an object is equal to that object's change in momentum, and occurs in the same direction
• Impulse = Δρ
• Formula: FΔt = mvf - mv₀

Collision example questions

• A 1500 kg car driving at 10 m/s colliding with a 2500 kg truck that was parked in a parking lot:
• Both experience the same magnitude force during the collision, due to Newton's Third Law
• Both experience the same magnitude of impulse during the collision
• Both experience the same change in momentum during the collision, due to the Impulse-Momentum Theorem
• The car experiences the greater acceleration during the collision, as dictated by Newton's Second Law

Ball & Wall example

• A ball moves to the right with a momentum of 3 kg⋅m/s, bounces off a wall, and then moves to the left with a momentum of 2 kg⋅m/s:
• The change in momentum of the ball = Δp = -5 kg⋅m/s
• The impulse that acted on the ball = -5 N⋅s
• If the ball has a mass of 100 g, its final velocity = -20 m/s to the left

Golf ball example

• A professional golf player can strike a golf ball, and it has a speed of 80 m/s after the strike; a typical golf ball has a mass of 45.93 g:
• Magnitude of the change in momentum of the golf ball during the strike = 3.67 kg⋅m/s
• The head of the golf club is in contact with a golf ball for typically no more than 0.5 ms; the average net force acting on the golf ball during the swing = 7,340 N

Concept questions

• Impulse and momentum are not the same; any moving object has a momentum, and impulse describes how the momentum changes when force is applied
• Two different objects are described as having experienced identical impulses, they do not necessarily experience the same magnitude of force; impulse is force multiplied by time, and if timespans were different, then the forces would be different to get the same impulse value, e.g., (5 N)(4 s) = (1 N)(20 s) = 20 N⋅s
• Two different objects are described as having experienced identical impulses, the impulses acting on them must point necessarily in the same direction; impulse is a vector, so to be identical, the direction must also be the same

Momentum of a system

• Impulse and momentum can be used to analyze what happens to a single object during an interaction, as well as analyze the system of objects involved in the collision
• The system (all objects involved in collision) will follow certain behaviors

Conservation of momentum

• When two objects collide, they exert a force on each other, perpendicular to their surfaces of contact
• Per Newton's Third Law, these forces are the same magnitude and opposite direction
• Each force causes each object to accelerate in the direction of the force
• A force pair causes an equal magnitude change in momentum in each object (equal magnitude impulses)
• The net change in momentum of the entire system of objects involved in the collision is zero
• Momentum is Conserved

Conservation of momentum formula

• The total linear momentum of a system of objects is constant, so long as there is no net external force acting on the system
• Formula: ptf = pto
• For a collision between two objects: m₁v₁f + m₂v₂f = m₁v₁₀ + m₂v₂₀
• vf = final velocity, vo = initial velocity
• Momentum is a vector, so the total momentum in both directions (x, y) must be conserved during a 2D collision

Types of collisions

• In all collisions where there is no external net force, momentum is conserved; there is more than one type of outcome when collisions occur:
• Elastic collisions conserve both momentum and energy; final velocity of the masses must be such that the total KE is unchanged in the system
• Inelastic collisions conserve only momentum; some energy is lost
• Perfectly Inelastic is a special case where objects stick together, in this collision, the most energy is lost

Box Collision example

• A 2 kg box slides across a frictionless surface at 5 m/s, colliding with a 4 kg box that had been at rest; after the collision, the 2 kg box slides at 1 m/s:
  • Initial momentum of the first box: p₁₀ = 10 kg⋅m/s
  • Initial momentum of the second box: p₂₀ = 0 kg⋅m/s
  • Initial momentum of the system: pto = 10 kg⋅m/s
  • Final momentum of the system: 10 kg⋅m/s (momentum is conserved; no net external forces acting)
  • Speed of the 4 kg box moving after the collision: v₂f = 2 m/s

Train Collision example

• An 80,000 kg train car moving at 5 m/s collides with, and couples to, a stationary 100,000 kg train car:
  • Final speed of the train cars after the collision: v = 2.2 m/s

Momentum in 2D collisions

• Momentum during a collision is still conserved during 2D motion, so long as there is no net external force acting on the system
• The amount of momentum along each direction (x and y directions) is the same before and after the collision