Unit 5: Momentum

📚Study Guide: Momentum

Unit 5: Momentum

Momentum provides a second powerful conservation law alongside energy, and for many collision and explosion problems it is the most direct route to a solution. Linear momentum, defined as the product of an object's mass and velocity (p = mv), is a vector quantity that captures both how massive an object is and how fast it is moving. The impulse-momentum theorem states that the net force acting over a time interval equals the change in momentum: J = F_net Δt = Δp. This theorem explains why airbags and crumple zones save lives—they increase the duration of a collision, which reduces the peak force for the same momentum change. The cornerstone of the unit is the law of conservation of linear momentum: if the net external force on a system is zero, the total momentum of that system remains constant. This holds true even when mechanical energy is not conserved, such as in perfectly inelastic collisions where objects stick together. You must distinguish between elastic collisions, where both momentum and kinetic energy are conserved, and inelastic collisions, where only momentum is conserved. In two-dimensional collisions, momentum is conserved independently along each perpendicular axis, requiring you to break velocities into components and write separate conservation equations for x and y. The unit also introduces the center of mass of a system, which moves as if all external forces were applied at that single point and all mass were concentrated there. For a two-object system, the center of mass velocity remains constant in the absence of external forces, a fact that often simplifies complex collision analyses. AP Exam questions frequently present ballistic pendulums, exploding projectiles, or skaters pushing off from each other, testing whether you can apply momentum conservation correctly while respecting its vector nature. Success depends on drawing clear before-and-after diagrams, labeling velocities as vectors, and never assuming a collision is elastic unless explicitly told so.

Key Concepts

• Linear Momentum: p = mv. A vector quantity describing the quantity of motion. Direction matters.
• Impulse-Momentum Theorem: The impulse J = F_net Δt equals the change in momentum Δp. A given momentum change can be achieved with a large force over a short time or a small force over a long time.
• Conservation of Momentum: If net external force on a system is zero, total momentum before equals total momentum after. This applies to collisions and explosions.
• Elastic vs. Inelastic Collisions: Elastic: momentum and kinetic energy conserved. Inelastic: only momentum conserved. Perfectly inelastic: objects stick together; momentum conserved, maximum KE lost.
• Two-Dimensional Collisions: Momentum is conserved separately in each perpendicular direction. Break all velocities into components before and after.
• Center of Mass: The weighted average position of mass in a system. The velocity of the center of mass is constant if no net external force acts.

Vocabulary

• Momentum (p): The product of mass and velocity. Vector. Unit: kg·m/s.
• Impulse (J): The product of average net force and the time interval over which it acts. Equal to change in momentum.
• Isolated System: A system on which no net external force acts, allowing momentum conservation.
• Elastic Collision: A collision in which both total momentum and total kinetic energy are conserved.
• Inelastic Collision: A collision in which total momentum is conserved but kinetic energy is not.
• Perfectly Inelastic Collision: A collision where the colliding objects stick together and move with a common final velocity.
• Center of Mass: The average location of the mass of a system, weighted by each particle's mass.

Essential Formulas

• p = m*v
• J = F_net * Δt = Δp
• m1*v1 + m2*v2 = m1*v1' + m2*v2' (conservation of momentum)
• v1 - v2 = -(v1' - v2') (relative velocity, elastic only)
• x_cm = (m1*x1 + m2*x2) / (m1 + m2)
• v_cm = (m1*v1 + m2*v2) / (m1 + m2)

Common Mistakes

• Assuming All Collisions Are Elastic: Unless the problem states "elastic" or gives information implying KE is conserved, assume the collision is inelastic and only momentum is conserved.
• Forgetting Momentum Is a Vector: In one dimension, this means careful sign management. In two dimensions, you must resolve into components.
• Applying Conservation When External Forces Act: If friction, gravity, or an applied force has a net effect during the collision, momentum is not conserved.
• Confusing Inelastic with Perfectly Inelastic: Inelastic simply means KE is lost. Perfectly inelastic means the objects stick together and share one final velocity.

AP Exam Strategies

• Draw Before-and-After Diagrams: Sketch the system immediately before and after the collision or explosion, labeling all masses and velocity vectors with directions.
• Write Momentum Conservation Separately for x and y: In 2D problems, do not try to combine directions. Write Σp_x(before) = Σp_x(after) and Σp_y(before) = Σp_y(after).
• Use the Relative Velocity Equation Only for Elastic Collisions: The equation v1 - v2 = -(v1' - v2') is derived from KE conservation and is invalid for inelastic collisions.
• Check if Kinetic Energy Is Conserved: Calculate total KE before and after. If they are equal, the collision is elastic. If not, it is inelastic.

Real-World Applications

• Car Crash Safety: Crumple zones and airbags increase the time Δt of a collision, reducing peak force on passengers for the same momentum change.
• Rocket Propulsion: Rockets expel mass backward at high speed. Conservation of momentum requires the rocket to move forward.
• Billiards: Nearly elastic collisions allow players to predict ball trajectories using momentum and energy conservation.