A positive charge of 4.90 μC is fixed in place. From a distance of 3.20 cm a from the fixed positive charge, a particle of mass 5.00 g and charge +3.20 μC is released from rest. What is the speed of the +3.20 μC when it is 7.50 cm away from the fixed positive charge?

I keep getting 1.005 m/s but can't figure out what I'm doing wrong!

Unknown

To solve this problem, you can use the principles of electrostatics and conservation of mechanical energy. Here are the steps to find the speed of the particle:

1. Determine the initial and final potential energies of the +3.20 μC charge.
- The initial potential energy can be calculated using the formula U_initial = (k * q1 * q2) / r_initial, where k is the electrostatic constant (k = 8.99 * 10^9 N m^2/C^2), q1 and q2 are the charges, and r_initial is the initial distance.
- In this case, the initial distance is 3.20 cm = 0.032 m. Plugging the values into the formula, U_initial = (8.99 * 10^9 N m^2/C^2) * (4.90 * 10^(-6) C) * (3.20 * 10^(-6) C) / 0.032 m.

2. Calculate the final potential energy of the +3.20 μC charge.
- The final potential energy can be found using the same formula as before, but with the final distance from the fixed charge, which is 7.50 cm = 0.075 m in this case. Plugging the values into the formula, U_final = (8.99 * 10^9 N m^2/C^2) * (4.90 * 10^(-6) C) * (3.20 * 10^(-6) C) / 0.075 m.

3. Apply the conservation of mechanical energy.
- According to conservation of mechanical energy, the change in potential energy is equal to the change in kinetic energy. In this case, the change in potential energy is U_final - U_initial.
- The change in kinetic energy (ΔK) is half of the mass of the particle multiplied by the square of its final velocity, ΔK = (1/2) * m * v_final^2.

4. Equate the change in potential energy to the change in kinetic energy and solve for the final velocity.
- Set U_final - U_initial = (1/2) * m * v_final^2 and rearrange the equation to solve for v_final.
- v_final = sqrt((2 * (U_final - U_initial)) / m)

Now, plug in the given values to find the final velocity:

- Plug in the values of U_initial, U_final, and m into the equation:
v_final = sqrt((2 * (U_final - U_initial)) / m)
v_final = sqrt((2 * ((8.99 * 10^9 N m^2/C^2) * (4.90 * 10^(-6) C) * (3.20 * 10^(-6) C) / 0.075 m - (8.99 * 10^9 N m^2/C^2) * (4.90 * 10^(-6) C) * (3.20 * 10^(-6) C) / 0.032 m)) / 0.005 kg)

Solving this equation using a calculator will give you the correct final velocity of the +3.20 μC charge when it is 7.50 cm away from the fixed positive charge.