Solving for Distance of Closest Approach for Moving Protons

[silver_gry]

Homework Statement

Two protons are moving directly toward one another. When they are very far apart, their initial speeds are 3.00*10^6 m/s. What is the distance of closest approach?

Homework Equations

The Attempt at a Solution

I really need help in question. I need to know how to start this question. Please help! Thank you!

 

[LowlyPion]

They both have kinetic energy.

If they have identical speeds, then they will both momentarily come to rest at their closest approach.

If they are at rest relative to each other then what is their potential energy with respect to each other at that moment?

 

[nlightner]

I would approach this problem by using principles from classical mechanics and electrostatics. The protons can be treated as point particles with positive charges, and their motion can be described using Newton's laws of motion. The distance of closest approach can be determined by considering the forces acting on the protons and the conservation of energy.

To start, we can use the given initial speeds and the mass of a proton (1.67*10^-27 kg) to calculate the kinetic energy of each proton. Since they are moving directly towards each other, we can assume that their initial velocities are equal in magnitude but opposite in direction. Therefore, the total initial kinetic energy of the system is given by:

KE = (1/2)*m*(3.00*10^6 m/s)^2 + (1/2)*m*(-3.00*10^6 m/s)^2 = 9*10^-12 J

Next, we can consider the forces acting on the protons. Since they are both positively charged, they will repel each other due to the electrostatic force. This force can be calculated using Coulomb's law:

F = k*(q1*q2)/r^2

where k is the Coulomb constant (8.99*10^9 N*m^2/C^2), q1 and q2 are the charges of the protons (equal to the elementary charge, 1.60*10^-19 C), and r is the distance between them.

At the distance of closest approach, the electrostatic force between the protons will be equal in magnitude to the centripetal force required to keep them in circular motion. This can be expressed as:

F = m*v^2/r

where m is the mass of a proton and v is the speed of each proton at the distance of closest approach.

Setting these two forces equal to each other, we can solve for r:

k*(q1*q2)/r^2 = m*v^2/r

r = k*(q1*q2)/(m*v^2)

Substituting in the values for k, q1, q2, m, and v, we get:

r = (8.99*10^9 N*m^2/C^2)*(1.60*10^-19 C)^2/(1.67*10^-27 kg)*(3.00*10^6 m/s)^2 =