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Suppose that two particles $P_1$ and $P_2$ interact with each other and that $P_2$ induces an instantaneous acceleration $a_{12}$ in $P_1$, while $P_1$ induces an instantaneous acceleration $a_{21}$ in $P_2$. Then

  1. these accelerations are opposite in direction and parallel to the straight line joining $P_1$ and $P_2$

2)the ratio of the magnitudes of these accelerations, $\frac{|a21|}{|a12|}$ is a constant independent of the nature of the mutual interaction between $P_1$ and $P_2$, and independent of the positions and velocities† of P1 and $P_2$.

For the first : I think induce means push

And for second one : The ratio of acceleration is constant. I think in this case both acceleration is constant so their ratio is constant.

But saying mutual interactions is odd I assume that means force. Now if mass is 0 and force is 0 then maybe the ratio will not depend but when they say it's velocity and position that is independent I can't agree on it since when you just tweak the position or velocity , the accleration will not be the same. So is my reasoning true or am I missing something?

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  • $\begingroup$ Write the Newton equation for each particle in the external force of the other one, and everything will follow from them. $\endgroup$ – Vladimir Kalitvianski Sep 10 '20 at 10:26
  • $\begingroup$ If two particles, each with volume, collide in two or three dimensions, I think its questionable to say the the accelerations are parallel to a line joining the two. Also an interaction with no force is not an interaction. $\endgroup$ – R.W. Bird Sep 10 '20 at 16:58
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I think mutual interaction means force on the particles in the system due to their internal properties like gravitational force due to mass , electromagnetic forces due to charge and not due to an external source.

If considering only gravitational force then , acceleration is induced not due to push . They are actually pulling each other and hence their acceleration have opposite direction and are on the line joining them since gravitational force acts along the line joining the masses. Taking the masses to be $M_1$ and $M_2$ the force on them will be

$F_{12} = \frac{GM_1M_2}{r^2}$ and similarly it can be written for the second particle.

And using Newton's second law of motion i.e.

$F =Ma$ ; you can calculate the ratio of the acceleration of the two particles. And this will be equal to the reciprocal of the ratio of their masses . Each time they come close force will change on them and thus acceleration of each particle will vary all along the distance between them but both will change by same amount and thus it will be constant.

If they were charged particles (with some mass ) then the force can be attractive as well as repulsive i.e. push as well as pull for different charge pairs. But again the forces are central and hence acceleration is parallel to the line joining them and in opposite directions and also their ratio can be calculated using the same technique.

Hope it helps 🙂.

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  • $\begingroup$ @Stupid question inc was that helpful ? $\endgroup$ – A student Sep 10 '20 at 11:26
  • $\begingroup$ Ah ok now I fully understand. $\endgroup$ – Stupid question inc Sep 10 '20 at 13:48

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