4.3 Conservation of Linear Momentum

Conservation of linear momentum means the total momentum of a system stays constant when no net external force acts on it. You use this idea to predict velocities before and after collisions and explosions by setting the total momentum before an interaction equal to the total momentum after.

Why This Matters for the AP Physics 1 Exam

Conservation of momentum is one of the most useful tools in mechanics because it lets you analyze interactions without knowing the messy details of the forces involved. On the AP Physics 1 exam, you can expect to:

• Set up momentum equations for collisions and explosions in one dimension and reason through two-dimensional setups.
• Connect momentum conservation back to Newton's third law and the impulse-momentum theorem.
• Reason about how changing a mass, speed, or angle changes the outcome without solving simultaneous equations.

This topic also supports experimental reasoning. Since the third free-response question is the Experimental Design and Analysis question, you may need to design a collision experiment, collect data, linearize it, and use a best-fit line to support a claim about momentum.

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Key Takeaways

• Total momentum of a system is the vector sum of each part's momentum: add magnitudes and directions, not just numbers.
• If the net external force on your chosen system is zero, total momentum stays constant.
• Momentum is conserved in every interaction; the system you pick decides whether you call a force internal or external.
• Internal forces come in equal and opposite pairs (Newton's third law), so they cannot change a system's total momentum.
• A nonzero net external force transfers momentum between the system and its surroundings, and that transfer equals the impulse.
• The center-of-mass velocity of a system stays constant when no net external force acts, even during internal collisions or explosions.

Center-of-Mass Velocity

The center of mass of a system represents the average position of all the mass, and it moves as if all the mass were concentrated at that point.

A group of objects, each with its own momentum, can be treated as a single system with one center-of-mass velocity that represents the overall motion.

• The center-of-mass velocity is found from: $\vec{v}_{cm} = \frac{\sum \vec{p}_i}{\sum m_i} = \frac{\sum m_i \vec{v}_i}{\sum m_i}$ where $\sum \vec{p}_i$ is the sum of all individual momenta and $\sum m_i$ is the total mass of the system.
• This treats the whole system as if its momentum were concentrated at one point.
• In the absence of a net external force, the center-of-mass velocity stays constant, no matter what happens internally.

Application: when a rocket in space ejects fuel, the center-of-mass velocity of the rocket-fuel system stays unchanged, even though the rocket itself accelerates opposite to the ejected fuel.

Total System Momentum

The total momentum of a system equals the vector sum of the momenta of all its parts. Because momentum is a vector, you have to track both magnitude and direction.

• In a collision between two objects, the total system momentum before equals the total system momentum after.
• This holds even when the individual momenta change dramatically.
• Written out: $\vec{p}_{total,before} = \vec{p}_{total,after}$

Application: when two billiard balls collide on a frictionless table, their individual momenta change, but the sum stays the same before and after.

Momentum Changes Within Systems

Internal forces between objects can shift individual momenta around while keeping the total momentum of the system fixed.

With no net external force, any change in one object's momentum must be balanced by an equal and opposite change in another object's momentum. This follows directly from Newton's third law.

• When two objects interact, the impulse object A exerts on object B is equal and opposite to the impulse B exerts on A.
• You can keep a system's total momentum constant by choosing system boundaries that include all interacting objects.
• If a system's total momentum does change, that change equals the impulse exerted on the system: $\vec{J} = \Delta \vec{p}$

Application: when a person jumps from a boat to a dock, the boat moves the opposite way. The person gains momentum one direction while the boat gains an equal amount the other direction, so the person-boat total stays constant.

System Selection for Momentum

Conservation in All Interactions

Momentum is conserved in every interaction, no matter which system you choose. Your choice of system changes how you interpret and calculate momentum changes, but not the underlying conservation principle.

• Defining clear system boundaries makes complex problems simpler.
• Expanding or shrinking the system can turn an external force into an internal force, or the reverse.

Zero Net External Force

When the net external force on your selected system is zero, the total momentum of that system stays constant. This works for both elastic and inelastic collisions.

• Written out: $p_{initial} = p_{final}$ or $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$
• This lets you find final velocities after collisions when you know the initial velocities and masses.
• In a perfectly inelastic collision, the objects stick together and move with one common final velocity, which you can find from momentum conservation.

Application: two ice skaters pushing off each other on frictionless ice move in opposite directions, with their momenta canceling out.

Nonzero Net External Force

If there is a nonzero net external force on your selected system, momentum is transferred between the system and its surroundings. The change in the system's momentum equals the impulse from that external force.

• When a ball bounces off the ground, the ground exerts an external force on the ball, transferring momentum between the ball and the Earth.
• Friction between moving objects and the ground is an external force that can change a system's total momentum over time.
• Expanding the system to include all interacting objects (like including the Earth) often restores momentum conservation.

How to Use This on the AP Physics 1 Exam

Problem Solving

• Define your system first, then check whether the net external force is zero. If it is, set total momentum before equal to total momentum after.
• Pick a positive direction and stick with it. Momentum is a vector, so leftward or downward values are negative.
• For a perfectly inelastic collision, the objects share one final velocity, so use $(m_1 + m_2)v_f$ on the final side.

Two-Dimensional Setups

• Split momentum into x and y components and conserve each direction separately.
• You are expected to set up the equations and reason qualitatively, not solve simultaneous equations.

Free Response

• For the Experimental Design and Analysis question, be ready to describe how you would collect collision data, linearize it, and use a best-fit line to support a claim.
• When asked to justify, connect equal and opposite impulses back to Newton's third law to explain why internal forces cannot change total momentum.

Common Misconceptions

• Momentum is not the same as kinetic energy. Momentum can be conserved in a collision even when kinetic energy is not, like in an inelastic collision.
• "Conserved" does not mean each object keeps its own momentum. It means the system total stays constant while individual momenta can change.
• Momentum is a vector. You cannot just add speeds or magnitudes; opposite directions partly or fully cancel.
• Conservation does not depend on the collision being elastic. Total momentum is conserved in elastic, inelastic, and perfectly inelastic collisions as long as the net external force is zero.
• An external force does not break the law of conservation. It transfers momentum to or from the system, equal to the impulse delivered.
• Whether momentum is "conserved" for a system depends on your system choice. Picking a wider system can turn an external force into an internal one.

Practice Problem 1: Conservation of Linear Momentum

Solution:

1. Identify the initial and final states:

   • Initial: Both skaters are stationary, so $v_{1i} = v_{2i} = 0$
   • Final: Second skater moves at $v_{2f} = 3$ m/s, and we need to find $v_{1f}$

2. Apply conservation of momentum:

   • Initial momentum: $p_i = m_1v_{1i} + m_2v_{2i} = 0$
   • Final momentum: $p_f = m_1v_{1f} + m_2v_{2f}$
   • Since momentum is conserved: $p_i = p_f$

3. Solve for the unknown velocity:

   • $0 = m_1v_{1f} + m_2v_{2f}$
   • $m_1v_{1f} = -m_2v_{2f}$
   • $v_{1f} = -\frac{m_2v_{2f}}{m_1}$
   • $v_{1f} = -\frac{40 \times 3}{60} = -2$ m/s

4. The negative sign indicates that the first skater moves in the opposite direction to the second skater, with a speed of 2 m/s.

Practice Problem 2: Center-of-Mass Velocity

Solution:

1. Identify the masses and velocities:

   • $m_A = 2$ kg, $v_A = 3$ m/s
   • $m_B = 4$ kg, $v_B = -2$ m/s
   • $m_C = 1$ kg, $v_C = 6$ m/s

2. Calculate the total momentum:

   • $p_{total} = m_Av_A + m_Bv_B + m_Cv_C$
   • $p_{total} = (2 \times 3) + (4 \times -2) + (1 \times 6)$
   • $p_{total} = 6 + (-8) + 6 = 4\ \text{kg}\cdot\text{m/s}$

3. Calculate the total mass:

   • $m_{total} = m_A + m_B + m_C = 2 + 4 + 1 = 7$ kg

4. Find the center-of-mass velocity:

   • $V_{cm} = \frac{\Sigma P}{\Sigma m} = \frac{p_{total}}{m_{total}} = \frac{4}{7} \approx 0.57$ m/s

5. The center of mass of the system moves at approximately 0.57 m/s in the positive x-direction.

Practice Problem 3: Two-Dimensional Momentum Setup

Solution:

1. Choose east as the +x-direction and north as the +y-direction.

2. Apply conservation of momentum separately in each direction:

   • x-direction: $m_1 v_{1i} = (m_1 + m_2) v_{xf}$
   • y-direction: $m_2 v_{2i} = (m_1 + m_2) v_{yf}$

3. The final velocity points somewhere northeast because the system has momentum in both x and y.

4. If the mass of Cart 2 increases while its initial speed stays the same, the y-component of momentum increases, so the final direction becomes more northward.

5. This kind of setup and qualitative reasoning is within AP Physics 1 scope; fully solving simultaneous two-dimensional momentum equations is not required.

Related AP Physics 1 Guides

• 4.1 Linear Momentum
• 4.2 Change in Momentum and Impulse
• 4.4 Elastic and Inelastic Collisions