Magnitude of radial component of jerk along line of centres

[tim_from_oz]

Homework Statement

Show that the component of the jerk along the line of centres of two gravitating bodies has magnitude 2G(m₁+m₂)r(dot)/r³.

Homework Equations

(Eq. 1) r(double dot) = - G(m₁+m₂)r(hat)/r² (acceleration of body 2 relative to body 1)

The Attempt at a Solution

Sorry for the (dots) and (hats), not sure how to put them on here.

Since I am only after the magnitude of the jerk radially between the two masses, and for each body this is equal to or opposite to Eq. 1, they can essentially be added, giving:

(Eq. 2) r(double dot) = 2G(m₁+m₂)r(hat)/r²

r(hat) is the unit vector, so this can be broken down into r(vector)/|r|, thus giving:

(Eq. 3) r(double dot) = 2G(m₁+m₂)r(vector)/r³

Finally, taking the derivative of the acceleration to get the jerk gives:

(Eq. 4) r(triple dot) = 2G(m₁+m₂)r(dot)/r³

This seems to have been too simple for me, so I think I've probably made some incorrect assumptions. I'm just looking for some guidance on whether I went down the right track or whether I should be heading down a different one.

This is an assessed problem, so just some guidance in the right direction would be appreciated as opposed to full solutions, etc.

Thanks.

 

[haruspex]

Finally, taking the derivative of the acceleration to get the jerk gives:

(Eq. 4) r(triple dot) = 2G(m1+m2)r(dot)/r3

Is the scalar r in the denominator not also a function of time?
(I feel that replacing the $\hat r$ with $\frac{\vec r}{r}$ was not helpful.)