How Do You Calculate the Initial Speeds of Two Electrons in a Collision?

[Millacol88]

Homework Statement

Two electrons separated by a large distance are fired directly at each other. The closest approach in this head-on collision is 4.0 * 10^-14 m. One electron starts with twice the speed of the other. Assuming there is no deflection from the original path, determine the initial speed of each electron.

Homework Equations

E_e = kq₁q₂ / r
E_k = 1/2 mv²

The Attempt at a Solution

I'm really lost as to how to start this. It must have something to do with electrical and kinetic energy but other than that I am mystified as to how to solve this.

 

[technician]

My first step would be to calculate the potential of 2 electrons at the given separation (equation 1)
The total KE would then be equal to this as the next step

 

[Millacol88]

OK, so I'm getting the velocities as 1.0 * 10⁸ m/s and 5.0 * 10⁷ m/s. I'm pretty confident about this but the textbook only gives one answer (5.3 *7 10) m/s. There should be two velocities as an answer, so I think that must be an error. Is my answer correct?

 

[cupid.callin]

Homework Statement

Two electrons separated by a large distance are fired directly at each other. The closest approach in this head-on collision is 4.0 * 10^-14 m. One electron starts with twice the speed of the other. Assuming there is no deflection from the original path, determine the initial speed of each electron.

Homework Equations

E_e = kq₁q₂ / r
E_k = 1/2 mv²

The Attempt at a Solution

I'm really lost as to how to start this. It must have something to do with electrical and kinetic energy but other than that I am mystified as to how to solve this.

OK, so I'm getting the velocities as 1.0 * 10⁸ m/s and 5.0 * 10⁷ m/s. I'm pretty confident about this but the textbook only gives one answer (5.3 *7 10) m/s. There should be two velocities as an answer, so I think that must be an error. Is my answer correct?

I suppose yes, your answers are correct !

 

[asteriatic]

Don't worry, it's completely normal to feel confused at first when faced with a new problem. Let's break it down step by step.

First, we need to understand the concept of electrical and kinetic energy. Electrical energy is the potential energy that exists between two charged particles, in this case the two electrons. Kinetic energy, on the other hand, is the energy an object possesses due to its motion.

In this problem, the two electrons are fired directly at each other, meaning they have the same initial direction and will collide head-on. The closest approach (4.0 * 10^-14 m) is the point at which the two electrons are closest to each other.

Now, we know that one electron starts with twice the speed of the other. Let's call the speed of the first electron v1 and the speed of the second electron v2. Since the two electrons have the same mass, we can use the equation for kinetic energy (Ek = 1/2 mv^2) to set up an equation:

1/2 m (2v1)^2 = 1/2 m v2^2

Simplifying this, we get:

2v1^2 = v2^2

This means that the initial speed of the first electron (v1) is equal to the square root of half the speed of the second electron (v2).

Next, we can use the equation for electrical energy (Ee = kq1q2 / r) to find the potential energy between the two electrons at the closest approach. Since the electrons have the same charge, we can rewrite the equation as:

Ee = k (q^2) / r

We know that the distance between the two electrons at the closest approach is 4.0 * 10^-14 m, so we can plug this in for r.

Now, we need to find the charge of each electron (q). This can be calculated using the value for the elementary charge (e = 1.602 x 10^-19 C). Since both electrons have the same charge, we can set up an equation:

k (q^2) / 4.0 * 10^-14 = 1.602 x 10^-19

Solving for q, we get:

q = √(1.602 x 10^-19 / k * 4.0