AP Physics C Mechanics- 4.4 Elastic and Inelastic Collisions- Study Notes- New Syllabus
AP Physics C Mechanics- 4.4 Elastic and Inelastic Collisions – Study Notes
AP Physics C Mechanics- 4.4 Elastic and Inelastic Collisions – Study Notes – per latest Syllabus.
Key Concepts:
- Elastic Collisions and Kinetic Energy Conservation
- Inelastic Collisions , Perfectly Inelastic Collision and Energy Transformation
Elastic Collisions and Kinetic Energy Conservation
An elastic collision is an interaction between two or more objects in which both the total momentum and the total kinetic energy of the system remain constant before and after the collision.
Although the total kinetic energy of the system is conserved, the individual kinetic energies of the objects involved may change. That is, energy may be redistributed between the objects, but the total remains the same.
Conservation Laws in Elastic Collisions
Momentum Conservation:
\( \mathrm{m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}} \)
Kinetic Energy Conservation:
\( \mathrm{\dfrac{1}{2} m_1 v_{1i}^2 + \dfrac{1}{2} m_2 v_{2i}^2 = \dfrac{1}{2} m_1 v_{1f}^2 + \dfrac{1}{2} m_2 v_{2f}^2} \)
- Both quantities are conserved only when no energy is lost to deformation, sound, or heat.
- Elastic collisions typically occur in idealized systems (e.g., atoms, billiard balls, or ideal spring collisions).
Nature of Elastic Collisions:
- The total kinetic energy of the system remains the same before and after collision.
- The kinetic energy of each object may change — one object may gain kinetic energy while the other loses it.
- The collision duration is very short, so external forces (like gravity) are negligible during the interaction.
- Elastic collisions satisfy both momentum conservation and energy conservation simultaneously.
1D Elastic Collisions:
For a one-dimensional elastic collision between two objects:
\( \mathrm{v_{1i} – v_{2i} = -(v_{1f} – v_{2f})} \)
- This relation shows that the relative velocity of approach before collision equals the relative velocity of separation after collision.
Physical Meaning:
- Elastic collisions conserve both energy and momentum — a condition not satisfied by inelastic collisions.
- Even though the total energy of the system is unchanged, energy may transfer from one object to another.
Example:
Two carts collide elastically on a frictionless track. Cart A (\( \mathrm{m_1 = 2.0\,kg} \)) moves at \( \mathrm{3.0\,m/s} \) toward stationary Cart B (\( \mathrm{m_2 = 1.0\,kg} \)). Find the final velocities of both carts after collision.
▶️ Answer / Explanation
Step 1: Use the 1D elastic collision formulas:
\( \mathrm{v_{1f} = \dfrac{(m_1 – m_2)}{(m_1 + m_2)}v_{1i} + \dfrac{2m_2}{(m_1 + m_2)}v_{2i}} \)
\( \mathrm{v_{2f} = \dfrac{2m_1}{(m_1 + m_2)}v_{1i} + \dfrac{(m_2 – m_1)}{(m_1 + m_2)}v_{2i}} \)
Step 2: Substitute values:
\( \mathrm{v_{1f} = \dfrac{(2 – 1)}{(3)}(3.0) + \dfrac{2(1)}{(3)}(0)} = 1.0\,m/s} \)
\( \mathrm{v_{2f} = \dfrac{2(2)}{(3)}(3.0) + \dfrac{(1 – 2)}{(3)}(0)} = 4.0\,m/s} \)
Step 3: Verify kinetic energy conservation.
- Initial: \( \mathrm{K_i = \tfrac{1}{2}(2)(3^2) = 9\,J} \)
- Final: \( \mathrm{K_f = \tfrac{1}{2}(2)(1^2) + \tfrac{1}{2}(1)(4^2) = 1 + 8 = 9\,J} \)
Result: \( \mathrm{v_{1f} = 1.0\,m/s} \), \( \mathrm{v_{2f} = 4.0\,m/s} \) → Total momentum and total kinetic energy are conserved.
Inelastic Collisions and Energy Transformation
An inelastic collision is an interaction between two or more objects in which the total momentum of the system is conserved, but the total kinetic energy decreases after the collision. Some of the system’s initial kinetic energy is transformed into other forms of energy such as heat, sound, deformation, or internal energy due to nonconservative forces acting during the collision.
Conservation Laws in Inelastic Collisions
Momentum Conservation:
\( \mathrm{m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}} \)
Kinetic Energy Change:
\( \mathrm{K_i > K_f} \)
- Total momentum remains constant if no external forces act on the system.
- Total kinetic energy decreases — energy is converted into internal or thermal energy by nonconservative forces.
Nature of Inelastic Collisions:
- Momentum is conserved, but kinetic energy is not.
- Some kinetic energy is lost to deformation, sound, or heat due to nonconservative internal forces.
- The total energy of the system (kinetic + potential + internal) remains constant — only its form changes.
Perfectly Inelastic Collision:
- A perfectly inelastic collision is a special case in which the colliding objects stick together after impact and move with a common velocity.
- This type of collision results in the maximum possible loss of kinetic energy consistent with momentum conservation.
Equation for Perfectly Inelastic Collisions:
\( \mathrm{m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v_f} \)
Energy Transformation:
The loss in kinetic energy during an inelastic collision equals the amount of energy transformed into other forms:
\( \mathrm{\Delta E = K_i – K_f} \)
- \( \mathrm{\Delta E} \): energy converted into nonmechanical forms (heat, sound, deformation).
- The system still obeys the principle of energy conservation — the total energy (including transformed forms) is constant.
Example:
A \( \mathrm{0.5\,kg} \) ball moving at \( \mathrm{4.0\,m/s} \) collides head-on and sticks to a stationary \( \mathrm{0.5\,kg} \) ball. Find the common velocity after collision and the loss of kinetic energy.
▶️ Answer / Explanation
Step 1: Apply momentum conservation (perfectly inelastic collision):
\( \mathrm{m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v_f} \)
Substitute values:
\( \mathrm{(0.5)(4.0) + (0.5)(0) = (1.0)v_f} \Rightarrow v_f = 2.0\,m/s} \)
Step 2: Compute initial and final kinetic energy.
- Initial: \( \mathrm{K_i = \tfrac{1}{2}(0.5)(4.0)^2 = 4.0\,J} \)
- Final: \( \mathrm{K_f = \tfrac{1}{2}(1.0)(2.0)^2 = 2.0\,J} \)
Step 3: Calculate energy lost.
\( \mathrm{\Delta E = K_i – K_f = 4.0 – 2.0 = 2.0\,J} \)
Result: The objects move together at \( \mathrm{2.0\,m/s} \) after collision, and \( \mathrm{2.0\,J} \) of kinetic energy is transformed into heat and deformation energy.