Chapter 7: Momentum

Momentum is defined by vector, p , such that it is the same as the product of mass and velocity \Rightarrow p = mv

  • This is the linear momentum, being a vector.
  • Given the units being kg m/s
  • K=12mv2=12m(p/m)2=p2/2mp=γ(2mk)K=\frac{1}{2}mv_{2}=\frac{1}{2}m(p/m)_{2}=p_{2}/2m\Rightarrow p=\gamma (2mk)
  • F=limΔg>0Δp/Δt\sum F=lim_{\Delta g->0}\Delta p/\Delta t
  • By Newton’s 2nd law
  • This general form includes the Δp\Delta p resulting from Δm\Delta m and Δv\Delta v
  • Momentumtotal=Momentum_{total}= vector sum of each individual momenta for any x particles in the system
  • Any system that is not acted upon by an external force is an isolated system
  • Given Internal Forces – within the system
  • Given External Forces – from outside the system
  • Provided that external forces cannot change the total momentum of the system
  • F=limΔt>0Δp/Δt0pfpipf=piF=lim_{\Delta t->0}\Delta p/\Delta t\Rightarrow 0 p_{f}-p_{i}\Rightarrow p_{f}=p_{i}
  • If the system is isolated
  • Which the linear momentum is conserved (constant)


Elastic vs Inelastic Collision

Total kinetic energy of the system after the collision is equal to the total kinetic energy before the collision \rightarrow \Rightarrow being Elastic

For the kinetic energy to be different of the initial and the final, it is Inelastic; If the objects stick together after colliding, the collision is Completely Inelastic


Impulse

Impulse, J, of a force is the product of the avg force, F, and the time interval J=FavgΔtJ=F_{avg}\Delta t \Rightarrow being a vector quantity in the direction as F

Impulse – Momentum Theorum

Aavg=VfVi/ΔtA_{avg}=V_{f}-V_{i}/\Delta t

F = ma F=mvfmvi/Δt\Rightarrow F=mv_{f}-mv_{i}/\Delta t FΔt=mvfmviF\Delta t=mv_{f}-mv_{i}

J=ΔPJ=\Delta P


Center of Mass

The center of mass is a point that represents the average location for the total mass of the system.

For Xcm=(m1x1+m2x2)/(m1+m2)X_{cm}=(m_{1}x_{1}+m_{2}x_{2})/(m_{1}+m_{2}) - Provides the velocity with a center of mass

Vcm=(m1v1+m2v2)/(m1+m2)V_{cm}=(m_{1}v_{1}+m_{2}v_{2})/(m_{1}+m_{2})


The 2 equations above can be dealt with more elements so we know the total momentum of any object(s) is the product of the total mass and the velocity of the center of mass.

  • P=p1+p2+...+pn(totalmVcenterofmass)P=p_{1}+p_{2}+...+p_{n}(total m \ast V_{center of mass})
  • P is conserved at any moment
  • Acceleration of center of mass:
  • Macm=mΛaΛ+mβaβ+...+mNaNMa_{cm}=m\Lambda a\Lambda +m\beta a \beta +...+m_{N}a_{N}


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