4.1 Linear Momentum

Linear momentum is mass times velocity, written as $\vec{p}=m\vec{v}$, and it points in the same direction as the velocity. It is a vector, so direction and sign matter, and it gives you a clean way to analyze collisions and explosions by comparing the system just before and just after an interaction.

Impulse, conservation of momentum, and collision problems later in Unit 4 all build on the definition of momentum. The experimental design and analysis free-response question can also use it, asking you to plan data collection, plot linearized graphs, and use a best-fit line to support a claim about a physical situation.

Momentum Is a Vector That Adds Across a System

• Momentum is defined as $\vec{p}=m\vec{v}$, with SI units of $\text{kg}\cdot\text{m/s}$.
• Momentum is a vector and always points in the same direction as the velocity.
• A heavy slow object and a light fast object can have the same momentum.
• The momentum of a system is the vector sum of the momenta of all its parts, so opposite momenta can cancel.
• A collision is a model where the forces between objects are much larger than the net external force during the interaction.
• An explosion is a model where internal forces push parts of the system apart.

Linear Momentum Description

Definition of Linear Momentum

Linear momentum represents the "quantity of motion" an object has, combining both its mass and velocity.

• Linear momentum ($\vec{p}$) equals an object's mass ($m$) times its velocity ($\vec{v}$): $\vec{p}=m\vec{v}$
• The SI unit for momentum is $\text{kg}\cdot\text{m/s}$ (kilogram-meter per second).
• A heavy object moving slowly can have the same momentum as a light object moving quickly.
• The greater an object's mass, the harder it is to change its velocity; the change in momentum equals the impulse applied.

Direction of Momentum

Momentum is a vector quantity, so it has both magnitude and direction, and it follows the same direction as velocity.

• Momentum always points in the same direction as the object's velocity.
• If velocity changes direction, such as after bouncing off a wall, momentum changes direction too.
• Two objects with equal mass but opposite velocities have equal and opposite momenta.
• The sign of momentum (positive or negative) depends on the coordinate system you choose for the problem.

Momentum of a System

The linear momentum of a system is the vector sum of the momenta of all objects in it:

$\vec{p}_{\text{system}}=\sum_i m_i\vec{v}_i$

Because momentum is a vector, the directions of the individual momenta matter when you add them. For example, two objects with equal and opposite momenta give zero total system momentum.

Momentum in Collisions and Explosions

Collisions and explosions are the situations where momentum analysis is especially useful.

• A collision is a model for an interaction in which the forces between the objects in the system are much larger than the net external force on those objects during the interaction.
  • Because the interaction time is very short, external forces are often negligible compared with the internal interaction forces.
• An explosion is a model for an interaction in which forces internal to the system move objects within that system apart.
  • The objects may start together and then separate because of those internal forces.
  • A firecracker going off, a rifle firing a bullet, or a person jumping from a boat are all examples that apply this model.
• In collision analysis, you usually focus only on the initial state just before the interaction and the final state just after it.
  • Because only the initial and final states are analyzed, you can use the object model to represent the colliding objects.
  • This makes momentum a powerful tool for analyzing interactions where the detailed forces are hard to track.

Choosing a Direction and Adding Momentum Vectors

Problem Solving:

• Always start by choosing a positive direction. Momentum's sign comes from that choice, so label it before plugging in numbers.
• When working with a system, add momenta as vectors. In two dimensions, handle the $x$ and $y$ components separately.
• Treat each object with the object model so you compare states before and after, instead of tracking forces during the interaction.

Common Trap:

• Do not drop direction. A magnitude alone is an incomplete answer for a vector quantity.
• Watch the difference between speed and velocity. A speed change from $3\ \text{m/s}$ to $-2\ \text{m/s}$ is a momentum change that uses signs, not just the size of the numbers.

Practice Problem 1: Linear Momentum Calculation

Solution:

For the first player (75 kg moving east at 5 m/s): $\vec{p}_1 = m_1\vec{v}_1 = (75 \text{ kg})(5 \text{ m/s east})$ $\vec{p}_1 = 375 \text{ kg}\cdot\text{m/s east}$

For the second player (90 kg moving west at 4 m/s): If we define east as the positive direction, then west is negative. $\vec{p}_2 = m_2\vec{v}_2 = (90 \text{ kg})(-4 \text{ m/s})$ $\vec{p}_2 = -360 \text{ kg}\cdot\text{m/s} = 360 \text{ kg}\cdot\text{m/s west}$

Practice Problem 2: Momentum Direction

Solution:

Initial momentum: $\vec{p}_i = m\vec{v}_i = (2 \text{ kg})(3 \text{ m/s}) = 6 \text{ kg}\cdot\text{m/s}$

Final momentum: $\vec{p}_f = m\vec{v}_f = (2 \text{ kg})(-2 \text{ m/s}) = -4 \text{ kg}\cdot\text{m/s}$

Change in momentum: $\Delta\vec{p} = \vec{p}_f - \vec{p}_i = -4 \text{ kg}\cdot\text{m/s} - 6 \text{ kg}\cdot\text{m/s} = -10 \text{ kg}\cdot\text{m/s}$

The negative sign indicates the change in momentum is in the negative x-direction. The magnitude of the change is $10\ \text{kg}\cdot\text{m/s}$.

Common Misconceptions

• Misconception: Momentum is the same as kinetic energy.
  Actually: Momentum is not the same as kinetic energy. Momentum is a vector ($\vec{p}=m\vec{v}$), while kinetic energy is a scalar that depends on speed squared.
• Misconception: A larger mass automatically means larger momentum.
  Actually: A larger mass does not automatically mean larger momentum. A light object moving fast can have more momentum than a heavy object barely moving.
• Misconception: Momentum points in the direction of force or acceleration.
  Actually: Momentum does not point in the direction of force or acceleration. It points in the direction of velocity.
• Misconception: The sign of momentum is built into the object.
  Actually: The sign of momentum is not built into the object. It comes from the coordinate direction you pick.
• Misconception: In an explosion, momentum is created from nothing.
  Actually: In an explosion, momentum is not created from nothing. Internal forces redistribute momentum within the system; they do not generate net momentum on their own.

Related AP Physics C: Mechanics Guides

• 4.3 Conservation of Linear Momentum
• 4.4 Elastic and Inelastic Collisions
• 4.2 Change in Momentum and Impulse