Two protons are moving directly toward one another. When they are very far apart, their initial speeds are 1.5e6 m/s. What is the distance of closest approach?

PLEASE HELP! im very confused.

It will be at a value of the separation distance R such that the initial kinetic energy, 2*(1/2)M V^2, is equal to the potential energy k e^2/R.

Note that you have to add the kinetic energies of the two particles, but the potential energy only has to be counted for bringing one particle within a distance R of the other.

R = ke^2/[M V^2]

k is the Coulomb constant (which you may need to look up) and e is the proton charge (same as the electron charge, except for sign). M is the proton mass, and V = 1.5*10^6 m/s

To determine the distance of closest approach between two protons moving directly towards each other, you can use the concept of conservation of energy.

The total mechanical energy of the system remains constant throughout the motion, which consists of the sum of kinetic energy and potential energy. At any point in their motion, the total energy is given by:

E = K + U

where:
E is the total mechanical energy
K is the kinetic energy
U is the potential energy

At the closest approach, all of the initial kinetic energy will be converted into potential energy due to mutual electrostatic repulsion. Thus, the total mechanical energy (E) will be equal to the potential energy (U) at this point.

The potential energy between two protons can be calculated using Coulomb's law:

U = (k * q1 * q2) / r

where:
U is the potential energy
k is the Coulomb's constant (8.99e9 N * m² / C²)
q1 and q2 are the charges of the protons (q1 = q2 = 1.6e-19 C)
r is the distance between the protons

Setting the total mechanical energy (E) equal to the potential energy (U), we have:

E = U

Substituting the values and rearranging the equation, we get:

K = (k * q1 * q2) / r

The kinetic energy (K) can be calculated using the equation:

K = (1/2) * mass * velocity²

where:
mass is the mass of a proton (1.67e-27 kg)
velocity is the initial speed of the protons (1.5e6 m/s)

Substituting the values and rearranging the equation, we have:

r = (k * q1 * q2) / (mass * (velocity/2)²)

Now we can calculate the distance of closest approach (r).

Let's plug in the values into the equation:

r = (8.99e9 N * m² / C²) * (1.6e-19 C)² / (1.67e-27 kg * (1.5e6 m/s / 2)²)

Calculating this expression will give you the distance of closest approach between the two protons.