Conservation of Momentum

Benjamin Crowell · Simple Nature · Ch. 2

Momentum p = mv is conserved in every isolated system. Impulse (force times time) changes momentum. Collisions split into elastic (KE conserved) and inelastic (KE lost to heat/deformation). The center of mass moves at constant velocity regardless.

Momentum

Momentum is mass times velocity: p = mv. It is a vector quantity. A truck moving slowly can have the same momentum as a bullet moving fast. The total momentum of an isolated system never changes.

Scheme

; Momentum: p = m * v
; Conservation: total momentum before = total momentum after

(define (momentum m v) (* m v))

; Two balls collide head-on
(define m1 3)  ; kg
(define v1 4)  ; m/s (rightward = positive)
(define m2 1)  ; kg
(define v2 -2) ; m/s (leftward = negative)

(define p-total (+ (momentum m1 v1) (momentum m2 v2)))

(display "p1 = ") (display (momentum m1 v1)) (display " kg m/s") (newline)
(display "p2 = ") (display (momentum m2 v2)) (display " kg m/s") (newline)
(display "Total momentum = ") (display p-total) (display " kg m/s") (newline)
(display "This total is conserved no matter what happens in the collision.")

Impulse

Impulse J = F * dt equals the change in momentum. A large force over a short time (baseball bat) and a small force over a long time (airbag) can produce the same impulse. The airbag spreads the force over more time, reducing peak force on your body.

Scheme

; Impulse: J = F * dt = delta p
; Same momentum change, different force/time

(define delta-p 600) ; kg m/s (change in momentum)

; Crash without airbag: 0.05 s
(define dt-no-bag 0.05)
(define F-no-bag (/ delta-p dt-no-bag))

; Crash with airbag: 0.5 s
(define dt-bag 0.5)
(define F-bag (/ delta-p dt-bag))

(display "Without airbag: F = ") (display F-no-bag) (display " N") (newline)
(display "With airbag:    F = ") (display F-bag) (display " N") (newline)
(display "Force ratio: ") (display (/ F-no-bag F-bag)) (display "x")

Elastic and inelastic collisions

In an elastic collision, both momentum and kinetic energy are conserved. In a perfectly inelastic collision, the objects stick together: momentum is conserved but kinetic energy is lost to deformation and heat. Most real collisions fall between these extremes.

Scheme

; Elastic collision: 1D, two masses
; Conservation of momentum AND kinetic energy
; v1' = (m1-m2)/(m1+m2) * v1 + 2*m2/(m1+m2) * v2
; v2' = 2*m1/(m1+m2) * v1 + (m2-m1)/(m1+m2) * v2

(define (elastic-collision m1 v1 m2 v2)
  (let ((total (+ m1 m2)))
    (list
      (+ (* (/ (- m1 m2) total) v1) (* (/ (* 2 m2) total) v2))
      (+ (* (/ (* 2 m1) total) v1) (* (/ (- m2 m1) total) v2)))))

; Equal masses, one at rest: the classic billiard shot
(define result (elastic-collision 1 5 1 0))
(display "Equal masses, billiard shot:") (newline)
(display "  v1' = ") (display (car result)) (display " m/s") (newline)
(display "  v2' = ") (display (car (cdr result))) (display " m/s") (newline)
; Ball 1 stops, ball 2 takes all the velocity

; Perfectly inelastic: objects stick together
(define (inelastic-collision m1 v1 m2 v2)
  (/ (+ (* m1 v1) (* m2 v2)) (+ m1 m2)))

(define v-stuck (inelastic-collision 3 4 1 0))
(display "Inelastic (3kg@4m/s + 1kg@0): v = ")
(display v-stuck) (display " m/s")

Center of mass

The center of mass of a system moves at constant velocity when no external forces act. Individual pieces can bounce, spin, and fragment, but the center of mass glides on serenely. This is why conservation of momentum is really about the center of mass.

Scheme

; Center of mass: x_cm = (m1*x1 + m2*x2) / (m1 + m2)
; v_cm = (m1*v1 + m2*v2) / (m1 + m2)

(define m1 3) (define v1 4)
(define m2 1) (define v2 -6)

(define v-cm (/ (+ (* m1 v1) (* m2 v2)) (+ m1 m2)))
(display "v_cm = ") (display v-cm) (display " m/s") (newline)
(display "v_cm is the same before and after any collision.") (newline)

; After perfectly inelastic collision, both move at v_cm
(define v-stuck (/ (+ (* m1 v1) (* m2 v2)) (+ m1 m2)))
(display "v_stuck = ") (display v-stuck) (display " m/s") (newline)
(display "v_stuck = v_cm? ") (display (= v-stuck v-cm))

Neighbors

← Ch. 1: Conservation of Energy

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