Systems of Particles and Linear Momentum: every key term you need (+ practice quiz)

More AP Physics C guides

• Kinematics
• Newton's Laws of Motion
• Work, Energy and Power
• Rotational Motion
• Gravitation

Linear Momentum

p = mv; a vector pointing along the velocity; the quantity Newton's second law changes: F = dp/dt.

Newton's Second Law (Momentum Form)

F_net = dp/dt; net force equals the time rate of change of momentum; reduces to ma for constant mass.

Impulse

J = ∫F dt = Δp; the integral of force over time equals the change in momentum; a vector measured in N·s.

Impulse-Momentum Theorem

J = Δp = m v_f − m v_i; the net impulse on an object equals its change in momentum.

Impulse from a Force Graph

Impulse equals the area under the force-time graph; a large force over a short time can equal a small force over a long time.

Average Force

F_avg = Δp/Δt; the constant force that would produce the same impulse over the collision time.

Conservation of Momentum

With no net external force, total momentum p_total is constant; internal forces cancel by Newton's third law.

Center of Mass

r_cm = (Σm_i r_i)/(Σm_i); the mass-weighted average position; moves as if all mass and external force acted there.

Center of Mass (Continuous)

x_cm = (1/M)∫x dm; for extended bodies, integrate position weighted by mass element dm.

Velocity of the Center of Mass

v_cm = p_total/M_total; the CM moves at constant velocity when no net external force acts.

Elastic Collision

Both momentum and kinetic energy are conserved; objects generally bounce apart with no energy lost to deformation.

Inelastic Collision

Momentum is conserved but kinetic energy is not; some KE converts to heat, sound, or deformation.

Perfectly Inelastic Collision

Objects stick together and move with a common final velocity; the maximum kinetic energy consistent with momentum conservation is lost.

Perfectly Inelastic Result

v_f = (m₁v₁ + m₂v₂)/(m₁ + m₂); common velocity after two objects stick together.

1D Elastic Collision Relative Speed

For a 1D elastic collision, v₁ − v₂ = −(v₁' − v₂'); the relative velocity reverses in sign.

Equal-Mass Elastic Collision

When a moving mass elastically strikes an equal stationary mass, they exchange velocities.

2D Collisions

Conserve momentum independently along each axis: Σpₓ and Σp_y are each conserved.

Recoil

When one object is pushed forward, another recoils so total momentum stays zero (e.g. a gun and bullet, or rocket exhaust).

Explosions

Internal forces separate a system; total momentum is conserved, so fragment momenta sum to the initial momentum.

Kinetic Energy in Terms of Momentum

KE = p²/(2m); relates kinetic energy to momentum magnitude and mass, useful in collision analysis.

External vs Internal Forces

Only external forces change a system's total momentum; internal (action-reaction) forces cancel in pairs.

Impulse as an Integral

J = ∫F dt = Δp; the area under an F-t graph; for a varying force the average force is J/Δt.

Center of Mass (Continuous)

x_cm = (1/M)∫x dm; for a rod with density λ(x), dm = λ dx and M = ∫λ dx.

Center of Mass Velocity

v_cm = Σm_iv_i/M = p_total/M; unchanged in any collision with no external impulse.

Elastic Collision Relative Speed

In a 1D elastic collision the relative speed of approach equals the relative speed of separation: v₁ − v₂ = −(v₁′ − v₂′).

Elastic Collision Formulas

v₁′ = [(m₁−m₂)v₁ + 2m₂v₂]/(m₁+m₂); equal masses exchange velocities.

Perfectly Inelastic Energy Loss

Fraction of KE lost when m₁ hits stationary m₂ and sticks is m₂/(m₁+m₂).

Ballistic Pendulum

Momentum conservation for the embed, then energy conservation for the swing: v = [(m+M)/m]√(2gh).

2D Collision Analysis

Conserve momentum in x and y separately; an elastic collision of equal masses (one at rest) yields a 90° separation.

Variable Mass: Rocket Equation

v − v₀ = u ln(m₀/m); thrust = u|dm/dt| where u is exhaust speed relative to the rocket.

Chain or Sand Onto Belt

Force needed to accelerate added mass to speed v is F = v(dm/dt), even at constant speed.

Momentum in CM Frame

Total momentum is zero in the center-of-mass frame; energy loss in a collision is easiest to see there.

Impulse Approximation

During a brief collision, large impulsive forces dominate; gravity's impulse over the short interval is neglected.

Explosions

Internal forces conserve momentum; released energy converts to kinetic energy shared inversely with mass.

Coefficient of Restitution

e = separation speed/approach speed; e = 1 elastic, e = 0 perfectly inelastic.