Newton's Law of Gravitation — Question 7

Question 7

Relative acceleration includes motion of both gravitating bodies

Two isolated point masses $M$ and $m$ are released from rest a distance $r_0$ apart. Let $r$ be their separation, initially along one fixed line.

1. Find $\ddot r$ initially and a short-time approximation for $r(t)$.

2. If $M=4m$, find how their individual displacements toward one another divide the decrease in separation.

Question 7 - Solution

Relative acceleration includes motion of both gravitating bodies

(a) Derivation and calculation

Choose $r=x_m-x_M>0$. The two accelerations give $$\ddot x_m=-\frac{GM}{r^2},\quad \ddot x_M=\frac{Gm}{r^2},\quad \boxed{\ddot r(0)=-\frac{G(M+m)}{r_0^2}}.$$ With initial relative speed zero, $$\boxed{r(t)\simeq r_0-\frac{G(M+m)}{2r_0^2}t^2},\qquad \frac{G(M+m)t^2}{r_0^3}\ll1.$$

(b) Result and interpretation

Let $\delta=r_0-r$ and let $s_M,s_m$ be positive inward displacements. Then $$s_M+s_m=\delta,\qquad Ms_M=ms_m.$$ For $M=4m$, $$\boxed{s_M=\delta/5,\qquad s_m=4\delta/5}.$$ The approximation must include the motion of both bodies.

Physical check

Both inward displacements are positive and their mass-weighted signed sum is zero.