Linear Momentum and Collision

Linear Momentum and Collision

• This chapter delves into the linear momentum of particles, the law of conservation of linear momentum in specific conditions
• It employs the conservation of energy to analyze translational motion during particle collisions.

Linear Momentum

• Linear momentum is the product of a system's mass (m) and its linear velocity (v).
• This represents the effect of force, fully grasped when mass and velocity are known.
• Momentum is a vector quantity, expressed as p = mv.
• In scalar notation, momentum is P = mv.
• Newton's second law of motion describes the rate of change of momentum being directly proportional to the applied force.
• Expressed as Fnet = dp/dt.
• Fnet = ma, shows that the relations are equivalent in classical mechanics and expressions of Newton's second law of motion.
• Linear momentum directly correlates with an object's mass and velocity.

SI Units and Dimensions

• SI Unit for momentum is kg/s.
• Dimension is MLT⁻¹.
• In two dimensions, it is represented as p = pxî + pyĵ
• px = mvx and py = mvy .

Impulse and Change of Momentum

• Impulse measures the effect of a large force acting briefly, when magnitude or time is hard to measure.
• Impulsive forces are great forces acting briefly, measured by rate of momentum change.
• Examples: kicking a ball, striking a nail, collision of bodies.
• Impulsive force varies from zero to maximum and back.
• Impulse (J) is the measure of total force effect, and product of force (F) and time (t).
• This is mathematically given J = Ft.
• Impulse is a vector with the same direction as the impacting force.
• An impulsive force F acts for a time t on mass m, changing velocity from vi to vf, with constant acceleration a.
• Utilizing Newton's second law (F=ma) and substituting a = (vf-vi)/t, results in Ft = m(vf - vi) or Ft = mvf - mvi.
• Ft = pf - pi or Ft = Δp, thus J = Δp.
• Impulse equals the change in momentum, described by the impulse momentum theorem.
• Both impulse and momentum are vectors, sharing units and dimensions.
• In collisions, a force F varies with time t.
• Area under the force vs. time graph represents the impulse or change in momentum.
• The area under the graph is J = ∫(from ti to tf) F dt = Δp.

Conservation of Linear Momentum

• When no external force acts on a system, the total linear momentum is constant.
• p = constant.
• The law of conservation of linear momentum is Σpi = Σpf.
• For particles A & B with masses m1 & m2, and velocities u1 & u2 before impact, and v1 & v2 after impact
• The law of conservation of linear momentum states m1u1 + m2u2 = m1v1 + m2v2

Recoil

• Practical instances is recoiling when firing a bullet from a gun
• The bullet gains forward momentum from gunpowder explosion.
• The gun attains equal backward momentum, termed recoil.
• If M = gun's mass, m = shot's mass, v = shot's velocity, and V = gun's velocity
• Then momentum is 0 = mv + MV and recoil velocity V = -mv/M, with "–" denoting recoiling direction.

Collisions

• Collisions are defined as isolated events where strong forces act on colliding particles briefly.
• Collisions should have clearly separated before and after phases.
• Examples of collisions include car accidents, billiard balls, or a hammer and nail.

Types of Collisions

• Elastic Collision: Both momentum and kinetic energy are conserved. Forces involved are conservative, like in atomic particle collisions.
• Inelastic Collision: Momentum of the system is conserved, but kinetic energy isn't.
• Examples: collisions between billiard balls or cars.
• Perfectly Inelastic Collision: Bodies stick together after impact or share velocity.
• Kinetic energy loss is maximum, but momentum remains conserved.

Elastic Collisions in One Dimension

• If m1 and m2 are masses, u1 and u2 initial velocities, and v1 and v2 final velocities, then:
• Total momentum before = Total momentum after, so m1u1 + m2u2 = m1v1 + m2v2.
• For elastic collisions, total kinetic energy is also conserved: 1/2 m1u1² + 1/2 m2u2² = 1/2 m1v1² + 1/2 m2v2².

Inelastic Collisions in One Dimension

• Ratio of relative velocity (post-collision) to relative velocity (pre-collision) is a constant and opposite in sign.
• (v1 - v2) / (u1 - u2) = -e where e is the coefficient of restitution.
• Therefore, v1 - v2 = -e(u1 - u2).
• If e = 1, the collision is perfectly elastic.
• If e = 0, the collision is perfectly inelastic.
• Generally, 0