Collisions in Systems of Particles (AP Physics C: Mechanics, Unit 4)

Collision

A short-time interaction in which two objects exert large forces on each other, changing their velocities.

External impulse

Impulse from forces outside the chosen system; often negligible during the brief collision interval, enabling momentum conservation for the system.

Isolated system (during collision)

An approximate model where net external impulse is negligible, so the system’s total momentum remains constant during impact.

Elastic collision

A collision in which both total linear momentum and total kinetic energy of the system are conserved.

Linear momentum

A vector quantity defined by (\vec p=m\vec v); total momentum is conserved when net external impulse is negligible.

Conservation of linear momentum

For two objects: (m1\vec v{1i}+m2\vec v{2i}=m1\vec v{1f}+m2\vec v{2f}); a vector equation (use components in 2D).

Kinetic energy (translational)

A scalar measure of motion energy: (K=\tfrac12 mv^2), where (v) is speed (magnitude of velocity).

Conservation of kinetic energy (collision)

Condition for an elastic collision: (\tfrac12 m1 v{1i}^2+\tfrac12 m2 v{2i}^2=\tfrac12 m1 v{1f}^2+\tfrac12 m2 v{2f}^2).

Velocity vs. speed

Velocity is a signed/vector quantity (includes direction); speed is the magnitude of velocity and is always nonnegative.

Vector equation (momentum)

An equation that must hold in each component; in 2D momentum conservation becomes separate x- and y-component equations.

Component form (2D momentum)

Writing momentum conservation as two equations: one for x and one for y (e.g., (p{x,i}=p{x,f}) and (p{y,i}=p{y,f})).

1D elastic collision

An elastic collision where all motion lies along one line; momentum and kinetic energy give two equations for two unknown final velocities.

Relative-speed reversal (1D elastic property)

In 1D elastic collisions, relative speed of approach equals relative speed of separation with opposite sign: (v{1i}-v{2i}=-(v{1f}-v{2f})).

Closed-form 1D elastic results

Formulas giving (v{1f}) and (v{2f}) in terms of masses and initial velocities, obtained by combining momentum conservation with the relative-speed relation.

Sign convention (1D collisions)

A consistent choice of positive direction so that velocities can be positive or negative; dropping signs is a common source of errors.

2D elastic collision

An elastic collision where motion can occur in a plane; momentum conservation must be applied in both x and y, plus kinetic energy conservation.

Underdetermined system (2D collisions)

In many 2D collisions there are more unknown velocity components than available equations (two from momentum + one from energy), so extra information is needed.

Scattering angle

The direction (angle) an object’s velocity makes after a collision; often provided to supply the extra information needed in 2D problems.

Line of impact

The direction along which the impulsive collision force effectively acts; constraints along this line can provide additional equations in collision problems.

Equal-mass, target-at-rest perpendicularity

In a 2D elastic collision with (m1=m2) and object 2 initially at rest, the final velocity vectors are perpendicular.

Dot product (perpendicular test)

A vector operation where (\vec a\cdot\vec b=0) indicates the vectors are perpendicular; used to show (\vec v{1f}\perp\vec v{2f}) in the special equal-mass case.

Inelastic collision

A collision in which total kinetic energy of the colliding objects is not conserved, though momentum is still conserved if external impulse is negligible.

Perfectly inelastic collision

An extreme inelastic collision where objects stick together and move with a common final velocity after impact.

Sticking-collision final velocity

For perfectly inelastic collisions: (\vec vf=\dfrac{m1\vec v{1i}+m2\vec v{2i}}{m1+m_2}), i.e., the center-of-mass velocity.

Coefficient of restitution (e)

A measure of “bounciness” along the line of impact: (e=\dfrac{\text{relative speed of separation}}{\text{relative speed of approach}}); in 1D, (e=\dfrac{v{2f}-v{1f}}{v{1i}-v{2i}}), with (e=1) elastic and (e=0) perfectly inelastic.