Speed of two charged, insulated spheres?

[XianForce]

Homework Statement

Two insulating spheres have radii 0.300 cm and 0.500 cm, masses 0.450 kg and 0.700 kg, and uniformly distributed charges of -2.00 µC and 3.00 µC. They are released from rest when their centers are separated by 1.00 m.
(a) How fast will each be moving when they collide? (Hint: Consider conservation of energy and of linear momentum.)

Homework Equations

F = k_eq₁q₂/r²
v = at + v₀
Δs = v₀t + .5at²

The Attempt at a Solution

I found the electrostatic force on the objects. Since they have opposite charges, I know that they move towards each other. I found their accelerations by dividing by their respective masses, and then plugged those into some of the kinematics equations to find the final velocity.

I can see that my methods/assumptions are wrong, I'm just not exactly sure about how to go about this problem.

 

[szynkasz]

Acceleration isn't constant as force increases while spheres get closer

 

[XianForce]

Acceleration isn't constant as force increases while spheres get closer

Yes, but then how do I solve it?

 

[szynkasz]

Your hint says you should use the law of conservation energy and momentum.

 

[Nickie]

I would approach this problem by first considering the forces acting on the two spheres. The electrostatic force between the two spheres can be calculated using Coulomb's law, which states that the force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. In this case, the force is attractive since the two spheres have opposite charges.

Next, I would consider the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred from one form to another. In this case, as the two spheres move towards each other, their potential energy decreases and is converted into kinetic energy. By equating the initial potential energy (at a separation of 1.00 m) to the final kinetic energy (at the point of collision), we can determine the final velocities of the two spheres.

I would also consider the principle of conservation of linear momentum, which states that the total momentum of a system remains constant unless acted upon by an external force. In this case, the total momentum of the system is zero at the point of release since the two spheres are at rest. As they move towards each other and collide, their momenta will be equal and opposite, resulting in a total momentum of zero once again.

Using these principles, I would set up equations for the conservation of energy and conservation of linear momentum and solve for the final velocities of the two spheres when they collide. This approach takes into account the masses, charges, and distances of the two spheres, as well as the forces acting on them, and provides a more comprehensive and accurate solution to the problem.