Energy and Momentum

Momentum and Impulse

• Linear Momentum: Product of an object's mass and velocity. $\vec{p} = m\vec{v}$
• Impulse: Change in linear momentum. $\,\vec{F}\Delta t = \Delta \vec{p}$
• Impulse is the net force acting over a time interval. $\,\vec{F}\Delta t = \vec{p}<em>f - \vec{p}</em>i$
• Impulse is also given as: $\,\vec{F}\Delta t = m\vec{v}<em>f - m\vec{v}</em>i$

Conservation of Momentum

• Law of Conservation of Momentum: The total linear momentum of a system remains constant in the absence of external forces. $\,\Delta \vec{p} = 0$
• In a system of two objects: $\,\vec{p}<em>{i1} + \vec{p}</em>{i2} = \vec{p}<em>{f1} + \vec{p}</em>{f2}$
• Which expands to: $m<em>1\vec{v}</em>{i1} + m<em>2\vec{v}</em>{i2} = m<em>1\vec{v}</em>{f1} + m<em>2\vec{v}</em>{f2}$

Two-Dimensional Collisions

• Momentum and velocities are resolved into two perpendicular components.
• The law of conservation of momentum applies independently in each direction.
• Equations:
  • $\,F<em>x\Delta t = \Delta p</em>x$
  • $\,F<em>y\Delta t = \Delta p</em>y$
  • $\,p<em>{i1x} + p</em>{i2x} = p<em>{f1x} + p</em>{f2x}$
  • $\,p<em>{i1y} + p</em>{i2y} = p<em>{f1y} + p</em>{f2y}$
• Non-perfectly Inelastic Collision: Objects don't stick, momentum is conserved, but kinetic energy is not.

Collisions

• Collisions can be analyzed using momentum and energy considerations.
• Perfectly Elastic Collisions: $\,m<em>1v</em>{i1} + m<em>2v</em>{i2} = m<em>1v</em>{f1} + m<em>2v</em>{f2}$
• Kinetic Energy Equation: $\,\frac{1}{2}m<em>1|\vec{v}</em>{i1}|^2 + \frac{1}{2}m<em>2|\vec{v}</em>{i2}|^2 = \frac{1}{2}m<em>1|\vec{v}</em>{f1}|^2 + \frac{1}{2}m<em>2|\vec{v}</em>{f2}|^2$
• Perfectly Inelastic Collisions: $\,\vec{v}<em>{f2} = \frac{m</em>1\vec{v}<em>{i1} + m</em>2\vec{v}<em>{i2}}{m</em>1 + m_2}$

Head-On Elastic Collisions

• Perfectly elastic head-on collisions conserve both momentum and kinetic energy.
• Equations:
  • $\,vec{v}<em>{f1} = (\frac{m</em>1 - m<em>2}{m</em>1 + m<em>2}) \vec{v}</em>{i1} + (\frac{2m<em>2}{m</em>1 + m<em>2}) \vec{v}</em>{i2}$
  • $\,vec{v}<em>{f2} = (\frac{m</em>2 - m<em>1}{m</em>1 + m<em>2}) \vec{v}</em>{i2} + (\frac{2m<em>1}{m</em>1 + m<em>2}) \vec{v}</em>{i1}$
• Special case where initial velocity of second object is zero:
  • $\,vec{v}<em>{f1} = (\frac{m</em>1 - m<em>2}{m</em>1 + m<em>2}) \vec{v}</em>{i1}$
  • $\,vec{v}<em>{f2} = (\frac{2m</em>1}{m<em>1 + m</em>2}) \vec{v}_{i1}$
• Special case where masses are equal:
  • $\,vec{v}<em>{f1} = \vec{v}</em>{i2}$
  • $\,vec{v}<em>{f2} = \vec{v}</em>{i1}$
• Special case where light object collides with stationary heavy object:
  • $\,vec{v}<em>{f1} ≈ -\vec{v}</em>{i1}$
  • $\,vec{v}_{f2} ≈ 0$

Work and Energy

• Work: Transfer of energy.
• Work Equation: $\,W = F \Delta d \cos\theta$
• Kinetic Energy: Energy of motion proportional to mass and square of velocity.
• Kinetic Energy Equation: $\,E_k = \frac{1}{2}mv^2$
• Work-Kinetic Energy Theorem: Work done is equal to the change in kinetic energy. $\,W = \Delta E_k$

Gravitational Potential Energy and Conservation of Energy

• Gravitational Potential Energy: Energy stored by an object's height relative to a reference level. $\,\Delta E_g = mg\Delta y$
• Law of Conservation of Energy: Energy can only be transformed or transferred. $\,E<em>T = E</em>k + E_g$
• For isolated systems: $\,E<em>{Ti} = E</em>{Tf}$
• Power: Rate at which energy is used or produced. $\,P = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$

Elastic Potential Energy and SHM

• Hooke’s Law: $F_x = -k\Delta x$
• Elastic Potential Energy: $\,E_e = \frac{1}{2}k|\Delta x|^2$
• Period of SHM (Mass on Spring): $\,T = 2\pi\sqrt{\frac{m}{k}}$

Conservation of Mechanical Energy (Revisited)

• Expanded equation: $\,E<em>T = E</em>k + E<em>g + E</em>e$
• For isolated systems: $\,E<em>{Ti} = E</em>{Tf}$
• Including work by non-conservative forces (open systems): $\,E<em>{Ti} = E</em>{Tf} + W_{nc}$