Force vs energy approach to a problem: Which is it? I have been working on a problem and I believe there should be two different answers to it, I used forces to calculate the required quantities while in another version used energies/potentials to solve for some quantities. I was having trouble reconciling the two approaches; such discrepancies also showed up when doing E&M problems, namely the book used an energy/momentum approach for moving particles whereas I opted for using forces.
Here is the problem:
Two particles of mass m and 11f are initially at rest an infinite distance apart.
Show that at any instant their relative velocity of approach attributable to gravitational attraction, is given by $\sqrt{2G\frac{m+M}{d}}$ where d is their separation at that instant.
This is What I did using forces:
-> If the separation of the particles is d at an instant after they start to move, I know the particles will be attracted with a force $F=G\frac{mM}{d^2}$, in the direction of the other particle, along the line joining them both.
->for particle of mass m: its acceleration, $a_m=F/m=F=G\frac{M}{d^2}$, if final velocity at this instant is $v_1$, its initial velocity, $u_1=0$ because it started from rest, and i am using s=d because that is the distance separating the particles(d is what they need to cover to approach each other) then from $v^2_1=u^2+2a_1d \implies v_1=\sqrt{2\frac{GM}{d}}$, $\rbrace$
->for particle of mass M: its acceleration, $a_M=F/M=F=G\frac{m}{d^2}$, if final velocity at this instant is $v_2$,  its initial velocity, $u_2=0$ because it started from rest then from $v^2_2=u^2+2a_1d \implies v_2=\sqrt{2\frac{Gm}{d}}$ $\rbrace$
This is what I Have found for the energy approach:
In this website:

Given the above two approaches I must ask;

*

*Which is deemed correct?


*Why are they different; shouldn't the two approaches yield the same answer?


*When using the force approach, did I make an invalid assumption somewhere?.


*Is there any way I can go ahead with the force approach and still get the same answer?
Some final thoughts:
1)I think the acceleration is not constant, so at each instant of the motion the accelerations will change due to the forces changing as the particles approach each other- Which means the equations of motions I used are invalid. But am I correct to dismiss the force approach only because of this? Are there any other ways in which it is possible to see that the force method should not be used in this problem?
I have been stuck with this for some time now, so would appreciate any new insight given.
 A: Both "forces" and "energy" approaches must give the same result. Always. They are mathematically equivalent.
Using one of the approach may be more convenient when solving some problems, for other problems the other approach may be more useful.
In this particular problem the distance between objects changes, so the forces and accelerations of the objects changes over time. Using "forces" approach is possible, but may be somewhat complicated. You need to write differential equation and then solve it.
"Energy" approach is very convenient because the total energy of the system is invariant, at always remains the same. Same for total momentum. You do not need to track all the details of what was happening to the system and how exactly objects were moving or interacting: information that energy and momentum of the system did not change allows to calculate required properties of the system without tracking what was happening to the system during the process.
I guess the mistake in your calculations is here: $v_1^2=u^2+2a_1d$
This formula ($v^2 - v_0^2 = 2as$) is correct only if the acceleration of the body is constant. But in this situation the acceleration changes all the time and this formula can not be used.
A: What you did using the equations of motion
$v^2_1=u^2+2a_1d \implies v_1=\sqrt{2\frac{GM}{d}}$
is not valid, as these need constant acceleration.
If you wanted to use a force approach you'd need integration
$v=\int a dt$ and since $a$ depends on $d$ that way would be quite tricky.
The energy way in green is easiest and the results with the blue bracket by them are ok, but shouldn't the second equation be
$v_2=\sqrt{2\frac{Gm^2}{d(m+M)}}$
giving a combined relative velocity of
$\sqrt{2\frac{Gm^2}{d(m+M)}}+\sqrt{2\frac{GM^2}{d(m+M)}}$
$=\sqrt{2\frac{G}{d(m+M)}}(m+M)$
leading to your first equation.
Hopefully this answers questions 1) and 3).  For 2) and 4), you are right that they should give the same result,  without too many details the force approach goes something like
$v=\int a dt = \int\frac{GM}{r^2}\times \frac{dt}{dr} dr $
$=\int \frac{GM}{r^2}\times \frac{1}{v} dr $
$v^2 = \int \frac{GM}{r^2} dr$
and this brings you back to the energy equations, so using energy is a lot easier.
