Jerk in Gravitation: Are Accelerations Equal in Cases of Varying Distances?

[nDever]

Lets suppose that we setup two instances involving two masses m1 and m2.

m1 > m2

Case 1. The two bodies are separated by a distance r = 20.
Case 2. The two bodies are separated by a distance r = 10.

Suppose that in both setups, we were to release all restraining forces on these two bodies at once and let gravity have its way. Both the accelerations will increase over time because r is decreasing but is the jerk of both cases equal?

 

[Drakkith]

What "Jerk" are you referring to?

 

[SHISHKABOB]

it's the rate of change of acceleration, or $\frac{d^{3}\vec{r}}{dt^{3}}$

 

[IsometricPion]

$$Gravitational\,\,acceleration:\quad\vec{a}_1=\frac{Gm_2\vec{r}}{|\vec{r}|^3}\Rightarrow\dot{\vec{a}}_1=\frac{Gm_2\dot{\vec{r}}}{|\vec{r}|^3}-\frac{3Gm_2 (\dot{ \vec{ r}} \cdot \vec{r}) \vec{r}}{| \vec{r}|^5} where\,\,\dot{q}=\frac{\partial{}q}{\partial{}t} for\,\,any \,\,quantity\,\,q$$This equation holds for all types of motion under only (two body) gravitational forces. Suppose the initial relative velocity of the two objects is zero, then near t=0 (when the two objects are released from rest) the motion is well approximated by uniform acceleration, for which the ratio v/r³ is approximately: $$|\frac{\dot{\vec{r}}}{|\vec{r}|^3}|\approx\frac{a_ot}{r_o^3}=\frac{Gm_2}{r_o^5}t$$So, the jerk will initially be zero and increase approximately linearly with time (over distances that are small when compared to the initial separation).

 

[PJC01]

I would say that the jerk, or the rate of change of acceleration, in these two cases is not necessarily equal. While the accelerations may increase over time in both cases due to the decreasing distance between the two masses, the initial accelerations will be different due to the differing masses of m1 and m2. This is because the force of gravity between two objects depends on their masses and the distance between them, as described by Newton's Law of Universal Gravitation. Therefore, the acceleration of m1 towards m2 will be greater than the acceleration of m2 towards m1, leading to a difference in jerk between the two cases.