What Is the Distance of Closest Approach in a Particle Collision?

[Tido611]

A proton and an alpha particle (q = +2.00e, m = 4.00u ) are fired directly toward each other from far away, each with an initial speed of 0.141c. What is their distance of closest approach, as measured between their centers? (Hint: There are two conserved quantities. Make use of both.)

I would figure that you could use conservation of energy in the sense that the energy of the system initially is the kinetic energies of the two particles combined (Enet1 = Kp + Ka). At the point of closest approach, their speeds should be zero, and hence Enet2 = Uelec = Kq1q2/r. From here it should be straightforward:

Enet1 = Enet2
Kp + Ka = Kq1q2/r

Then solve for "r".

However, this is incorrect. Perhaps my assumption that the alpha particle (4 times the mass, 2 times the charge) stops completely is wrong. At this point, I really have no idea.

 

[NeoDevin]

It might work better if you set it up in a frame where the center of mass is stationary.

Then use the conservation of energy and momentum.

 

[m2003]

I would first like to commend you for taking the time to think critically about this problem and for seeking help in finding a solution. This is a great approach to problem-solving and shows your dedication to understanding the concepts at hand.

In regards to your approach, you are correct in using conservation of energy in this scenario. However, the mistake lies in assuming that the alpha particle will stop completely at the point of closest approach. In reality, both the proton and alpha particle will experience a change in direction and continue moving after the closest approach.

To solve this problem, we can use the conservation of momentum and the conservation of energy equations. The conservation of momentum equation states that the total momentum before the collision is equal to the total momentum after the collision. In this case, the initial momentum is zero since both particles are initially at rest. The final momentum can be calculated using the masses and velocities of the particles after the closest approach.

The conservation of energy equation can be used to find the distance of closest approach. As you mentioned, the initial energy of the system is the sum of the kinetic energies of the particles. At the point of closest approach, all of this energy is converted to potential energy due to the electrostatic force between the particles. Therefore, we can equate the initial kinetic energy to the final potential energy and solve for the distance of closest approach.

I encourage you to try solving the problem using these equations and see if you can arrive at the correct answer. If you need further assistance, don't hesitate to seek help from a teacher or a fellow scientist. Remember, collaboration and critical thinking are key components of the scientific process. Good luck!