Two electrons (each with mass m = 9.1 × 10−31 kg and charge q = −1.6 × 10−19 C) are located along the circum- ference of an imaginary circle with radius R = 95 cm. One is fixed in place at the top of the circle (i.e., the 12 o’clock position), while the other is placed on the right side of the circle (i.e., the 3 o’clock position) and then released from rest. What is the maximum speed v that the second elec- tron will reach when it is infinitely far away from the first electron? The symbolic answer should be stated in terms of m, q, R, θ, and k, though not all of these quantities may be needed. Can you also plug in the numbers into the symbolic answer and solve for v.

The problem involves two electrons on the circumference of a circle. The first electron is fixed at the top (12 o'clock position), while the second electron is initially at the right side (3 o'clock position) and is released from rest. The goal is to find the maximum speed \( v \) of the second electron when it is infinitely far away from the first electron.

### Step 1: Energy Conservation

Since no external forces are acting on the system, we can use the conservation of mechanical energy. Initially, the second electron has potential energy due to the electrostatic interaction with the first electron and no kinetic energy because it starts from rest.

At infinite separation, the potential energy will be zero, and all the initial potential energy will have been converted into kinetic energy.

The total energy at the start (when the second electron is on the circle) is given by:

\[
E_{\text{initial}} = U_{\text{initial}} + K_{\text{initial}}
\]

Where:
- \( U_{\text{initial}} \) is the initial electrostatic potential energy between the two electrons.
- \( K_{\text{initial}} \) is the initial kinetic energy, which is zero since the second electron starts from rest.

At infinity, the potential energy is zero and the second electron will have a maximum kinetic energy:

\[
E_{\text{final}} = U_{\text{final}} + K_{\text{final}}
\]

Where:
- \( U_{\text{final}} = 0 \) (since the potential energy is zero at infinite separation).
- \( K_{\text{final}} = \frac{1}{2}mv^2 \), where \( v \) is the final speed of the electron.

By conservation of energy:

\[
U_{\text{initial}} = \frac{1}{2}mv^2
\]

### Step 2: Expression for Initial Potential Energy

The potential energy between two charges separated by a distance \( r \) is given by Coulomb's law:

\[
U_{\text{initial}} = \frac{k q_1 q_2}{r}
\]

Here, \( q_1 = q_2 = q = -1.6 \times 10^{-19} \, \text{C} \), and the initial distance between the electrons is the radius of the circle, \( R = 0.95 \, \text{m} \).

So, the initial potential energy is:

\[
U_{\text{initial}} = \frac{k q^2}{R}
\]

### Step 3: Solving for Maximum Speed \( v \)

From energy conservation, we now have:

\[
\frac{k q^2}{R} = \frac{1}{2}mv^2
\]

Solving for \( v \):

\[
v = \sqrt{\frac{2k q^2}{mR}}
\]

This is the symbolic answer in terms of \( k \), \( q \), \( m \), and \( R \).

### Step 4: Plugging in the Values

Now, we'll substitute the known values:
- \( k = 8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2 \) (Coulomb constant),
- \( q = 1.6 \times 10^{-19} \, \text{C} \),
- \( m = 9.1 \times 10^{-31} \, \text{kg} \),
- \( R = 0.95 \, \text{m} \).

Let's plug these into the equation:

\[
v = \sqrt{\frac{2 \times (8.99 \times 10^9) \times (1.6 \times 10^{-19})^2}{9.1 \times 10^{-31} \times 0.95}}
\]

I'll calculate this value for you.The maximum speed \( v \) that the second electron will reach when it is infinitely far away from the first electron is approximately \( 23.07 \, \text{m/s} \).

Would you like further details or explanations on any part of this process?

Here are 5 questions related to the concept:
1. How would the result change if the charges were not identical?
2. What would happen if the initial separation was different?
3. How does the result differ for particles with different masses?
4. Can this method be applied to gravitational interactions instead of electrostatic ones?
5. What are the limitations of assuming energy conservation in this problem?

**Tip:** Always double-check units when applying physical formulas, as consistency in units ensures accurate results.

Two electrons (each with mass m = 9.1 × 10−31 kg and charge q = −1.6 × 10−19 C) are located along the circumference of an imaginary circle with radius R = 95 cm. One is fixed in place at the top of the circle (i.e., the 12 o’clock position), while the other is placed on the right side of the circle (i.e., the 3 o’clock position) and then released from rest. What is the maximum speed v that the second electron will reach when it is infinitely far away from the first electron? The symbolic answer should be stated in terms of m, q, R, θ, and k, though not all of these quantities may be needed. Can you also plug in the numbers into the symbolic answer and solve for v.

Explore how to calculate the maximum speed of an electron using Coulomb's law and energy conservation. This problem involves key concepts such as electrostatic potential energy, kinetic energy, and conservation of mechanical energy. Learn how to apply these formulas with given values for the charge and mass of an electron.

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