4.1 Linear Momentum
In classical mechanics, linear momentum represents "mass in motion." All objects have mass; so if an object is moving, then it has momentum. It is a fundamental vector quantity that points in the exact same direction as the object's velocity.
Linear Momentum (p): The product of an object's mass and its velocity.
Linear Momentum Formula
- p = momentum (kg·m/s)
- m = mass (kg)
- v = velocity (m/s)
⚠️ Remember: Momentum is a Vector!
Unlike kinetic energy (which is a scalar), momentum has direction. A ball thrown to the right at 5 m/s has a positive momentum. If it bounces back to the left at 5 m/s, its momentum is now negative. The signs matter immensely in calculations!
4.2 Change in Momentum and Impulse
To change an object's momentum, a net external force must be applied over a period of time. This application of force over time is called Impulse.
Impulse (J): The change in momentum of an object when it is subjected to a force over a time interval.
Impulse from a Graph
Forces during collisions (like a bat hitting a baseball) are rarely constant. They spike to a maximum value and drop back down to zero. To find the impulse in these realistic scenarios, we use graphical analysis.
The Area under a Force vs. Time (F-t) graph equals the Impulse (Change in Momentum).
Thought Experiment: Airbags and Egg Drops
When you drop an egg on a pillow versus dropping it on concrete, the egg's total change in momentum (Δp) is the same (it goes from falling at velocity 'v' to 0 m/s). Because Δp is constant, extending the time (Δt) of the collision using a pillow (or a car airbag) drastically reduces the average force (Favg) acting on the object.
Greater Δt = Smaller Favg
4.3 Conservation of Linear Momentum
One of the most powerful laws in physics is the Law of Conservation of Momentum. It allows us to predict the outcomes of collisions and explosions without needing to know the complex internal forces acting between the objects.
Conservation of Linear Momentum: If the net external force acting on a system is zero, the total momentum of the system remains constant.
Velocity of the Center of Mass
If momentum is conserved within a closed system, the velocity of the center of mass (vcm) of the system remains perfectly constant, regardless of how the individual pieces collide, stick together, or explode apart inside the system.
Center of Mass Velocity Formula
(Notice that this is just Total Momentum divided by Total Mass!)
4.4 Elastic and Inelastic Collisions
While total momentum is conserved in all isolated collisions, total kinetic energy is not. We categorize collisions based on what happens to the system's kinetic energy during the interaction.
[Image illustrating elastic, inelastic, and perfectly inelastic collisions between two objects]| Type of Collision | Momentum Conserved? | Kinetic Energy Conserved? | Distinguishing Feature |
|---|---|---|---|
| Elastic | ✅ Yes | ✅ Yes | Objects bounce off each other perfectly with no loss of energy. (Rare in the macro-world). |
| Inelastic | ✅ Yes | ❌ No | Objects bounce, but some kinetic energy is converted to thermal/sound energy (deformation). |
| Perfectly Inelastic | ✅ Yes | ❌ No (Max loss) | Objects stick together and move with a shared final velocity. |
Perfectly Inelastic Collision Equation
Unit 4 Key Takeaways
Momentum is a VECTOR. Pay close attention to positive and negative directions.
Impulse is the change in momentum (J = Δp) AND the area under an F-t graph.
Extending the time of a collision (Δt) reduces the average impact force.
If net external force = 0, the system's momentum is perfectly conserved.
The center of mass velocity of a closed system remains constant during a collision.
Elastic = Kinetic Energy Conserved. Inelastic = Kinetic Energy Lost.
End of Unit 4 Study Guide.