Energy and Momentum Conservation in One Dimension

1. Momentum

1.1. Definition

• Momentum (p) is a measure of the mass in motion. It is a vector quantity.
• Formula: $p = mv$
  • Where:
    • $p$ = momentum (kg m/s)
    • $m$ = mass (kg)
    • $v$ = velocity (m/s)
• Units: kg m/s

1.2. Change in Momentum

• Change in momentum ($\Delta p$) is given by: $\Delta p = m\Delta v = mv_f - mv_i$
  • Where:
    • $v_f$ = final velocity
    • $v_i$ = initial velocity

1.3. Impulse

• Impulse (J) is the change in momentum of an object. It is a vector quantity.
• Impulse is also equal to the force applied to an object multiplied by the time interval over which it acts.
• Formula: $J = F\Delta t = \Delta p$
  • Where:
    • $F$ = force (N)
    • $\Delta t$ = time interval (s)
• Units: Ns (Newton-seconds)
• Impulse can be determined from a force-time graph (area under the curve).

KEY TAKEAWAY: Momentum is a measure of mass in motion, and impulse is the change in momentum. They are both vector quantities.

2. Conservation of Momentum

2.1. Isolated System

• An isolated system is one where the net external force acting on the system is zero.

2.2. Law of Conservation of Momentum

• In an isolated system, the total momentum remains constant.
• The total momentum before an interaction (e.g., collision, explosion) is equal to the total momentum after the interaction.
• Formula: $\Sigma p_{initial} = \Sigma p_{final}$
• For a two-object system: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$
  • Where:
    • $m_1, m_2$ = masses of objects 1 and 2
    • $u_1, u_2$ = initial velocities of objects 1 and 2
    • $v_1, v_2$ = final velocities of objects 1 and 2

2.3. Types of Collisions

• Elastic Collision: A collision in which kinetic energy is conserved.
• Inelastic Collision: A collision in which kinetic energy is not conserved. Kinetic energy is transformed into other forms of energy such as heat, sound, or deformation.
• Perfectly Inelastic Collision: A collision in which the objects stick together after the collision. Kinetic energy is not conserved.

EXAM TIP: Remember that momentum is always conserved in an isolated system, regardless of the type of collision. Kinetic energy is only conserved in elastic collisions.

3. Energy

3.1. Kinetic Energy

• Kinetic energy (Ek) is the energy an object possesses due to its motion.
• Formula: $E_k = \frac{1}{2}mv^2$
  • Where:
    • $E_k$ = kinetic energy (J)
    • $m$ = mass (kg)
    • $v$ = velocity (m/s)
• Units: Joules (J)

3.2. Gravitational Potential Energy

• Gravitational potential energy (Eg) is the energy an object possesses due to its position in a gravitational field relative to a reference point (usually ground level).
• Formula: $E_g = mg\Delta h$
  • Where:
    • $E_g$ = gravitational potential energy (J)
    • $m$ = mass (kg)
    • $g$ = acceleration due to gravity (9.8 m/s²)
    • $\Delta h$ = change in height (m)
• Units: Joules (J)

3.3. Elastic Potential Energy

• Elastic potential energy (Es) is the energy stored in a deformed elastic object, such as a spring.
• Formula: $E_s = \frac{1}{2}k\Delta x^2$
  • Where:
    • $E_s$ = elastic potential energy (J)
    • $k$ = spring constant (N/m)
    • $\Delta x$ = extension or compression of the spring (m)
• Units: Joules (J)

3.4. Work Done

• Work (W) is the transfer of energy. It is a scalar quantity.
• Formula: $W = Fd\cos\theta$
  • Where:
    • $W$ = work done (J)
    • $F$ = force (N)
    • $d$ = displacement (m)
    • $\theta$ = angle between force and displacement. If force and displacement are in the same direction, $\cos\theta = 1$
• Units: Joules (J)
• Work done can also be found from the area under a force-distance graph.

3.5. Power

• Power (P) is the rate at which work is done or energy is transferred. It is a scalar quantity.
• Formula: $P = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$
  • Where:
    • $P$ = power (W)
    • $W$ = work done (J)
    • $\Delta t$ = time interval (s)
    • $\Delta E$ = change in energy (J)
• Alternative Formula: $P = Fv$
• Units: Watts (W) = J/s

COMMON MISTAKE: Students often confuse work and energy. Work is the transfer of energy, not energy itself.

4. Conservation of Energy

4.1. Law of Conservation of Energy

• Energy cannot be created or destroyed; it can only be transformed from one form to another or transferred from one object to another.
• In a closed system (no energy enters or leaves), the total energy remains constant.

4.2. Energy Transformations

• Energy can be transformed between kinetic, gravitational potential, elastic potential, heat, sound, light, and other forms.
• Examples:
  • A falling object: Gravitational potential energy is converted into kinetic energy.
  • A bouncing ball: Gravitational potential energy -> kinetic energy -> elastic potential energy (during compression) -> kinetic energy -> gravitational potential energy (some energy is lost as heat and sound).
  • A spring-mass system: Elastic potential energy is converted into kinetic energy and vice versa.

4.3. Energy Dissipation

• In real-world scenarios, some energy is often “lost” due to dissipative forces like friction and air resistance.
• This energy is usually converted into thermal energy (heat) or sound.
• Although this energy is “lost” from the mechanical system, it is still conserved within the larger system (e.g., the environment).

4.4. Applying Conservation of Energy

• To solve problems involving energy conservation:
  1. Identify the initial and final states of the system.
  2. Identify the forms of energy present in each state.
  3. Apply the law of conservation of energy: $E_{initial} = E_{final} + E_{dissipated}$
  4. Solve for the unknown quantity.

STUDY HINT: Practice drawing energy transformation diagrams to visualize how energy changes forms in different scenarios.

5. Combining Momentum and Energy Conservation

5.1. Elastic Collisions Revisited

• In an elastic collision, both momentum and kinetic energy are conserved.
• Equations:
  • Momentum conservation: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$
  • Kinetic energy conservation: $\frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2$
• These equations can be used to solve for the final velocities of the objects after the collision.

5.2. Inelastic Collisions Revisited

• In an inelastic collision, momentum is conserved, but kinetic energy is not.
• Only the momentum conservation equation can be used directly.
• The change in kinetic energy can be calculated to determine the amount of energy dissipated: $\Delta E_k = E_{k,final} - E_{k,initial}$

5.3. Perfectly Inelastic Collisions Revisited

• In a perfectly inelastic collision, the objects stick together after the collision, simplifying the equations.
• Momentum conservation: $m_1u_1 + m_2u_2 = (m_1 + m_2)v_f$
• Where $v_f$ is the final velocity of the combined mass.

VCAA FOCUS: VCAA loves to ask about the difference between elastic and inelastic collisions and how to apply the appropriate conservation laws. Expect questions requiring you to calculate energy losses or final velocities.

6. Work-Energy Theorem

6.1. Definition

• The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.
• Formula: $W_{net} = \Delta E_k = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

6.2. Applications

• The work-energy theorem provides an alternative way to analyze motion, especially when dealing with variable forces.
• It connects the concepts of work, energy, and motion.

REMEMBER: The work-energy theorem is a powerful tool for relating work done to changes in kinetic energy.

APPLICATION: Consider a car braking to a stop. The work done by the friction force from the brakes is equal to the change in the car’s kinetic energy.

7. Vertical Spring-Mass Systems

7.1. Equilibrium Position

• The equilibrium position of a vertical spring-mass system is where the spring force balances the gravitational force.
• At equilibrium: $k\Delta x = mg$

7.2. Energy Considerations

• When analyzing a vertical spring-mass system, consider the following forms of energy:
  • Kinetic energy ($E_k$)
  • Gravitational potential energy ($E_g$)
  • Elastic potential energy ($E_s$)

7.3. Conservation of Energy in Spring-Mass Systems

• The total mechanical energy of the system remains constant (in the absence of dissipative forces).
• $E_{total} = E_k + E_g + E_s = constant$
• Problems often involve finding the maximum compression or extension of the spring, or the velocity of the mass at a particular point.

EXAM TIP: When dealing with vertical spring-mass systems, carefully define your reference point for gravitational potential energy. It’s often easiest to set it at the spring’s initial uncompressed position.