Unit 4 Beta

Linear Momentum: the one-page sheet

4.1 Linear Momentum

Momentum p = mv is a vector along the velocity. A system's momentum is the vector sum of its parts, and its center of mass moves at v_cm = Σp/Σm. In collisions and explosions the internal forces dominate, so the momentum just before equals the momentum just after.

Momentum p = mv is a vector in the direction of the velocity, in kg·m/s.
A system's momentum is the vector sum of its parts. Opposite momenta cancel; kinetic energies never do.
Same momentum does not mean same kinetic energy: K = p²/(2m), so the lighter object has more K.
The velocity of the center of mass is v_cm = Σp/Σm.
The collision model: forces between colliding objects are internal and much larger than outside forces, so the system's momentum is the same just before and just after. An explosion is the same model in reverse.

it has momentum p = mv, a vector along v, in kg·m/s the system's momentum is the vector sum, and momenta in opposite directions cancel the result is the velocity of the center of mass, v_cm = Σp/Σm the internal forces dominate, so the system's momentum is the same just before and just after

momentum: The product of an object's mass and velocity, p = mv: a vector along the velocity, in kg·m/s.
collision model: The model of a brief, strong interaction (a collision or an explosion) in which internal forces far exceed external ones, so the system's momentum is the same just before and just after.
velocity of the center of mass: The velocity of a system's center of mass: the system's momentum divided by its total mass, v_cm = Σp/Σm.

4.2 Change in Momentum and Impulse

Impulse J = F_avg Δt is the area under a force-time graph, and it equals the change in momentum. A fixed change in momentum spread over more time needs less force. The slope of a momentum-time graph is the net force.

Impulse J = F_avg Δt, a vector along the force, in N·s (= kg·m/s). It is the area under a force-time graph.
The impulse-momentum theorem: J_net = Δp = m v_f − m v_i. Keep the signs: a bounce reverses v.
The same Δp over a longer time needs a smaller force: airbags, mats, bent knees.
F_net = Δp/Δt, the slope of a momentum-time graph. For constant mass this is Newton's second law, F_net = ma.

it delivers an impulse J = F_avg Δt, the area under the force-time graph, in N·s the object's momentum changes by exactly that amount: J = Δp a longer contact time means a smaller average force the slope of the p-t graph is the net force: F_net = Δp/Δt

impulse: The product of the average force and the time it acts, J = F_avg Δt: a vector in N·s, equal to the area under a force-time graph.
impulse-momentum theorem: The net impulse on an object equals its change in momentum: J = Δp.
force-time graph: A graph of force against time; the area under it is the impulse.
momentum-time graph: A graph of momentum against time; its slope is the net force on the object.

4.3 Conservation of Linear Momentum

Internal forces cancel in pairs, so only external forces change a system's momentum. With zero net external force, total momentum and the center of mass's velocity stay constant. Choosing the system decides which forces are external; always state the condition.

Internal forces come in third-law pairs, so their impulses cancel: they move momentum between parts but never change the total.
Conservation of momentum: when the net external force on a system is zero, its total momentum is constant: Σp_before = Σp_after.
The choice of system decides whether momentum is constant. A cart alone in a collision is not; both carts together are.
With no net external force, the center of mass moves at constant velocity, whatever the parts do.
Justify with the condition: "the net external force on the two-cart system is zero", never "because momentum is conserved".

the third law makes the forces, and their impulses, equal and opposite one object's momentum gain is the other's loss, so internal forces cannot change the total the system's total momentum, and its center of mass's velocity, stay constant its forces become external and the system's momentum can change

conservation of momentum: When the net external force on a system is zero, the system's total momentum stays constant.
choice of system: Deciding which objects are inside the system; it determines which forces are internal (cannot change the total momentum) and which are external (can).

4.4 Elastic and Inelastic Collisions

Collisions conserve momentum when no net external force acts. Elastic collisions also keep the total kinetic energy; inelastic ones lose some; perfectly inelastic ones stick and lose the most. Start every collision from momentum, and use energy only when the collision is elastic or after it is over.

Momentum first: in every collision with no net external force, Σp_before = Σp_after.
Elastic collision: total kinetic energy unchanged (individual K can change). Inelastic: total K decreases. Perfectly inelastic: objects stick, v_f = Σp/Σm.
Never use energy conservation to find the speed after a sticky collision. Use momentum, then use energy afterward if needed.
A momentum bar chart shows each object's momentum (with sign) before and after; the total bars match.
A bounce delivers more impulse than a stick: reversing momentum is a bigger change than stopping.

the system's momentum is the same before and after, whatever the type of collision the collision is elastic: total kinetic energy is unchanged, though it moves between the objects the collision is inelastic: total kinetic energy decreases the collision is perfectly inelastic: one common final velocity, the largest possible loss of kinetic energy

elastic collision: A collision in which the total kinetic energy of the system is the same before and after (momentum is also conserved).
inelastic collision: A collision in which momentum is conserved but the total kinetic energy decreases, the difference becoming thermal energy, sound and deformation.
perfectly inelastic collision: An inelastic collision in which the objects stick together and move with one common velocity, v_f = Σp/Σm.
momentum bar chart: Bars showing each object's momentum (with its sign) and the system total before and after an interaction; with no net external impulse, the totals match.