Final velocity of 2 spheres attracted by gravity and electrostatic charge

[Runaway]

Homework Statement

Two insulating spheres having radii 0.22 cm
and 0.42 cm, masses 0.16 kg and 0.47 kg, and
charges −4 μC and 5 μC are released from
rest when their centers are separated by 1 m.
How fast is the smaller sphere moving when
they collide?
Answer in units of m/s.

Homework Equations

F_g=(G * m₁ * m₂)/r²
F_e= (k q₁ * q₂)/r
F=ma
V_f²=V₀²+ 2a * X

The Attempt at a Solution

I'm AP Physics and the class is supposed to be algebra and trig. based, but I don't see any way to solve this problem without calculus because the forces will constantly increase as the two spheres become closer. So my question is: can this be done algebraically or does it have to be done with calculus, and if it can be done algebraically how do I calculate the sphere's velocity when the force acting upon it, and thus it's acceleration, is constantly increasing?
Work so far:
F= F_g + F_e
F= (k q₁ * q₂)/r + (G * m₁ * m₂)/r²
and using F=ma, a₁ = F / m₁ = (k q₁ * q₂)/r + (G * m₁ * m₂)/r² / m₁
and using V_f²=V₀²+ 2a * X
V_f for m₁=sqrt(V₀²+ 2a * X)=
sqrt(2 ( k * q₁ * q₂)/r + (G * m₁ * m₂)/r² / m₁)* X))
where X = the initial distance between the two spheres

 

[cupid.callin]

i will give you two concepts using which you can solve this ...
you can assume that net external force is 0
so "Total Internal Energy of the System" is conserved and also "Angular Momentum of the System" is constant

 

[Runaway]

I don't see how that helps me account for the fact that the acceleration isn't constant due to the forces increasing as the distance between them decreases.

 

[gneill]

The total energy is conserved, so PE is trading off with KE. The "experiment" begins with PE and no KE...

 

[eczeno]

and feel free to ignore gravity. it will have no effect.

 

[Runaway]

Thanks for all your help, I figured it out: 16.0m/s

 

[cupid.callin]

and feel free to ignore gravity. it will have no effect.

you mean gravity due to earth?
coz i guess gravity b/w 2 spheres will have effect

 

[eczeno]

well, you have that backward. the gravity between the two spheres will be insignificant compared to the electric force between them (like a factor of 10^10 less). they will both be affected by Earth's gravity, but this problem does not ask you to address that.