AP Physics C Mechanics- 4.1 Linear Momentum- Study Notes- New Syllabus
AP Physics C Mechanics- 4.1 Linear Momentum – Study Notes
AP Physics C Mechanics- 4.1 Linear Momentum – Study Notes – per latest Syllabus.
Key Concepts:
- Linear Momentum
- Application of Momentum
Linear Momentum
Linear momentum is the product of an object’s mass and its velocity. It describes the quantity of motion an object possesses and how difficult it is to stop or change that motion.
Relevant Equation:
\( \mathrm{\vec{p} = m\vec{v}} \)
- \( \mathrm{\vec{p}} \): linear momentum (unit: \( \mathrm{kg \cdot m/s} \))
- \( \mathrm{m} \): mass of the object (kg)
- \( \mathrm{\vec{v}} \): velocity vector of the object (m/s)
Nature of Momentum:
- Momentum is a vector quantity — it has both magnitude and direction.
- The direction of momentum is always the same as the direction of velocity.
- The greater the mass or velocity, the larger the momentum.
Physical Meaning:
- Momentum measures an object’s resistance to changes in motion when forces act.
- In collisions or interactions, momentum is often transferred between objects but the total momentum of an isolated system is conserved.
Units and Dimensions:
| Quantity | Symbol | SI Unit | Dimension |
|---|---|---|---|
| Momentum | \( \mathrm{\vec{p}} \) | \( \mathrm{kg \cdot m/s} \) | \( \mathrm{M L T^{-1}} \) |
Example:
A \( \mathrm{2.0\,kg} \) cart moves east at a velocity of \( \mathrm{3.0\,m/s} \). Find its linear momentum, and describe its direction.
▶️ Answer / Explanation
Step 1: Use the formula for linear momentum:
\( \mathrm{\vec{p} = m\vec{v}} \)
Step 2: Substitute values:
\( \mathrm{\vec{p} = (2.0)(3.0) = 6.0\,kg \cdot m/s} \)
Step 3: Indicate direction:
Since the velocity is east, the momentum is also directed east.
Final Result: \( \mathrm{\vec{p} = 6.0\,kg \cdot m/s \, \text{(east)}} \)
Application of Momentum
Momentum is a fundamental quantity used to analyze interactions between objects — particularly during collisions and explosions. In these short-duration events, internal forces between objects dominate, and the law of conservation of momentum allows accurate prediction of the system’s outcome without requiring detailed knowledge of the forces involved.
Why We Study Momentum:
- Momentum provides a powerful method for analyzing interactions where forces act over very short times and are difficult to measure directly.
- During these interactions:
- The total momentum of an isolated system remains constant.
- Internal forces between objects obey Newton’s third law and cancel within the system.
- This allows us to relate only the initial and final states of motion — simplifying the analysis.
Collisions:
A collision is an interaction between two or more objects during which they exert large internal forces on each other for a short time. These internal forces are much greater than any external forces acting on the system during the interaction.
- Because external forces are negligible, the system’s total momentum is conserved:
- This conservation principle allows determination of the final velocities of objects after collision without calculating the detailed forces or time of contact.
- The object model — treating objects as point masses — is used because only initial and final velocities matter, not their internal structure or shape.
Explosions:
An explosion is an interaction in which forces internal to the system push the objects within it apart. Although mechanical energy may increase due to the release of stored internal energy, the total momentum of the system remains conserved (if no external forces act).
- Explosions are momentum-conserving but energy-transforming interactions — internal chemical or elastic potential energy is converted into kinetic energy of separating parts.
- Example systems include fireworks, rockets, or a bullet fired from a gun.
| Interaction Type | System Forces | Momentum Conservation | Energy Change | Example |
|---|---|---|---|---|
| Collision | Large internal forces | Conserved (if isolated) | May increase or decrease (elastic vs. inelastic) | Billiard balls, car crash |
| Explosion | Internal forces push objects apart | Conserved (if isolated) | Increases (internal → kinetic) | Rocket launch, grenade explosion |