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Momentum and impulse

This article explores the concepts of momentum and impulse, including how momentum is affected by mass and velocity, and the principles of conservation of momentum. It also discusses different types of collisions, such as elastic and inelastic collisions, and provides strategies for solving problems involving collisions. Additionally, the article touches on rocket propulsion and the equation for calculating thrust.

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Momentum and impulse

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  1. Momentum and impulse

  2. The linear momentum - p of an object of mass m moving with velocity v is the product of its mass and velocity: p = mv SI unit: kg m/s • Doubling the mass of the velocity of an object double its momentum • Doubling both quantities, quadruples its momentum Momentum is a vector quantity ( quiz 6.1/161)

  3. Changing the momentum of an object require application of a force: Fnet = ma = m Δv/Δt = Δ(mv)/Δt The change in an object’s momentum Δp divided by the elapsed time Δt equals the constant net force Fnet acting on the object If a constant force acts on a object. The impulse I delivered to the object over a time interval Δt is given by: I = FΔt SI unit: kg m/s (ex 6.2/163)

  4. Conservation of momentum • A2 /188 • When a collision occurs in an isolated system, the total momentum of the system doesn’t change with the passage of time • The total momentum is conserved! • If we consider an isolated system (no external forces) of two particles before and after they collide

  5. F21Δt = m1v1f-m1v1i F12Δt = m2v2f-m2v2i F21= -F12 F21Δt= -F12Δt m1v1f-m1v1i = -(m2v2f-m2v2i) m1v1i +m2v2i= m1v1f+m2v2f When no net forces acts on the system, the total momentum remains ct. in time (conservation of momentum)

  6. Collisions • For any type of collision, the momentum of the system just before the collision= to the momentum after the collision as long as the system is isolated • Elastic collision- both momentum and kinetic energy are conserved • Inelastic collision- momentum is conserved but kinetic energy is not • Perfect inelastic collision, momentum is conserved, KE is not, and the 2 objects stick together after the collision (quiz 6.3/168)

  7. Perfectly inelastic collision: m1v1i +m2v2i =(m1+m2)vf vf=(m1v1i +m2v2i )/(m1+m2) (quiz 6.6/172)

  8. Elastic collision: m1v1i +m2v2i= m1v1f+m2v2f m1(v1i-v1f) = m2(v2f-v2i) 1/2m1v1i2+1/2 m2v2i2=1/2 m1v1f2=1/2m2v2f2 m1(v1i2-v1f2)=m2(v2f2-v2i2) m1(v1i-v1f)(v1i+v1f)= m2(v2f-v2i)(v2f+v2i) v1i+v1f =v2f+v2i v1i-v2i = -(v1f-v2f)

  9. One dimensional collision; pb. Strategies • Choose coordinate axis (along direction of motion) • Diagram • Conservation of momentum: write total momentum before and after • Conservation of energy: write the expression (for elastic or perfect inelastic) • Solve (ex. Page 182-188)

  10. Glancing collisions

  11. For a general collision in 3D, the conservation of momentum principle implies that total momentum of the system in each direction is conserved m1v1ix +m2v2x= m1v1fx+m2v2fx m1v1i y+m2v2iy= m1v1fy+m2v2fy Glancing collision: if an object with mass m1 collides with a mass m2 that is initial at rest, after collision, obj.1 moves at an angleθ with horizontal, and obj.2 moves at an angleΦ with horizontal

  12. x component: m1v1i + 0 = m1v1f cosθ+m2v2f cosΦ y component: 0+ 0 = m1v1f sinθ+m2v2f sinΦ Conservation of energy: 1/2m1v1i2+0 =1/2 m1v1f2+1/2m2v2f2 (if the collision is inelastic, the KE of the system is not conserved, the 3rd equation does not apply)

  13. Two dimensional collision; pb. Strategies • Choose coordinate axis (use both x, and y- coordinates) • Diagram • Conservation of momentum: write total momentum before and after • Conservation of energy: write the expression (for elastic or perfect inelastic) • Solve

  14. Rocket Propulsion (page 177)

  15. Equating the total initial momentum of the system with the total final momentum: (M + Δm)v =M(v+ Δv) + Δm(v-ve) ve –fuel ejected speed M Δv = - veΔm but Δm =- ΔM M Δv = - veΔM (calculus:) vf-vi =veln(Mi/Mf)

  16. The thrust of the rocket = the force everted on the rocket by the ejected exhaust gases instantaneous thrust= Ma = M Δv/Δt = |veΔM/Δt|

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