Momentum

Linear Momentum

• Product of mass and linear velocity
• Symbol is p; units are kg·m/s
• p = mv
• Vector whose direction is same as velocity
• Related to inertia and kinetic energy
• Large momentum due to large mass or high speed; no velocity--no momentum

Impulse

• Net force can change velocity and momentum
• F _netma = mDv/Dt; so F_netDt = mDv
• Product of force and time interval is impulse
• Impulse also equals change in momentum due to force
• Units are Ns which equals kg·m/s
• Must have average or constant force for equation
• If force is not constant, impulse found by area under force vs. time graph
• To increase momentum change due to force, increase time force is applied
• To decrease force in collision, increase time of impact

Conservation of Momentum

• If no external force acts, and mass doesn't change, then momentum can't change
• Total vector sum of momentum is constant if no external forces act on closed system.
• Internal forces between objects within system have no effect on total momentum
• Momentum can be transferred between objects, but sum remains constant.

Collisions

• Isolated event in which a strong force acts on two or more bodies for a short time.
• Momentum is transferred, but conserved
• Two types of collisions, inelastic and elastic
• Most real collisions are at least partially inelastic

Inelastic Collisions

• When objects stick together after colliding and/or significant deformation, sound, light are produced
• Since objects stick together, only one final velocity
• m ₁v₁ + m₂v₂ = (m₁ + m₂)v_f
• Energy not conserved

Elastic Collisions

• Objects rebound off each other
• No significant deformation, sound, light, etc.
• Kinetic energy and momentum conserved
• Have two initial and final velocities
• m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f (Cons. of Momentum)
• ½ m₁v_1i² + ½ m₂v_2i² = ½ m₁v_1f² + ½ m₂v_2f²
• Only true elastic collisions are gas molecules

Two Dimensional Collisions

• Must use vectors to figure momentum
• In elastic collision of two bodies of equal mass, one moving, one stationary, kinetic energy equation simplifies to Pythagorean theorem
• Therefore, angle between objects is right angle, and we can use trig functions to find resultant velocity

Angular Momentum

• Using Newton's 2nd law for rotation, T = Ia = IDw/Dt; so TDt = IDw
• thus angular impulse equals change in angular momentum
• Symbol for angular momentum is L and units are kgm²/s; L = Iw
• Angular momentum is always conserved if no external torques act on the system.
• Example: flywheel in car engine
• If rotational inertia is changed by redistributing mass, angular velocity changes to keep angular momentum constant
• Example: spinning ice skater moves arms to change speed of spin

Summary

• Linear momentum, p = mv
• A change in momentum due to a force acting over a time is impulse FDt = mDv
• Inelastic collisions have objects sticking together, energy losses due to deformation
• Elastic collisions have objects bouncing off each other, kinetic energy is conserved
• As long as no external forces act, and no mass enters or leaves system, momentum is conserved in all collisions
• Angular momentum L = Iw is momentum due to rotation.
• Angular impulse is change in angular momentum due to external torque applied over time: TDt = IDw
• Angular momentum is conserved if no external torques act on system
• A change in rotational inertia will change angular speed because momentum is constant
• To solve collision problems, set total initial momentum equal to total final momentum

Vocabulary

• Linear momentum
• Angular momentum
• Impulse
• Angular impulse
• Elastic collision
• Inelastic collision
• Law of conservation of momentum