Solving for Distance of Closest Approach for Moving Protons

• silver_gry
In summary, two protons with an initial speed of 3.00*10^6 m/s are moving directly towards each other. At their closest approach, they will both momentarily come to rest if they have identical speeds. If they are at rest relative to each other, their potential energy with respect to each other at that moment is zero.
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?

The Attempt at a Solution

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

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?

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 =

1. What is the distance of closest approach for moving protons?

The distance of closest approach for moving protons is the shortest distance between two protons as they move towards each other. It is the distance at which the protons will experience the strongest repulsive force due to their positive charges.

2. How is the distance of closest approach calculated for moving protons?

The distance of closest approach is calculated using the Coulomb's law, which states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. In the case of moving protons, their velocities and directions of motion are also taken into account in the calculation.

3. What factors affect the distance of closest approach for moving protons?

The distance of closest approach for moving protons is affected by the velocities of the protons, their charges, and the angle at which they approach each other. The distance also depends on the medium in which the protons are moving, as the medium can affect their velocities and hence the distance of closest approach.

4. Why is the distance of closest approach important in nuclear collisions?

The distance of closest approach is important in nuclear collisions because it determines the probability of two protons colliding and undergoing a nuclear reaction. A smaller distance of closest approach increases the likelihood of a collision, while a larger distance reduces the chances of a reaction occurring.

5. Can the distance of closest approach be measured experimentally?

Yes, the distance of closest approach can be measured experimentally using particle accelerators and detectors. By measuring the trajectories of the protons and their angles of deflection, the distance of closest approach can be calculated and compared with the theoretical value determined from the Coulomb's law.

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