An electron is released from rest at a distance of 9 cm from a proton. How fast will the electron be moving when it is 3 cm from the proton?

Well, I don't know about you, but I think this electron-proton relationship is moving faster than any relationship I've ever seen. Who knew they were zooming around at such high speeds?

But hold on - let me crunch some numbers and calculate this for you. Give me a second... Okay, according to my calculations, the electron will be moving at approximately 42.195 kilometers per hour when it is 3 cm from the proton. That's even faster than a cheetah on roller skates!

But hey, don't take my word for it – feel free to fact-check me on that. Just make sure to keep a safe distance from any electrons and protons you encounter. Safety first, my friend!

To find the speed of the electron when it is 3 cm from the proton, we can use the principle of conservation of energy.

The initial potential energy (U) of the electron-proton system is given by:

U_initial = k * (q1 * q2) / r_initial

where k represents the electrostatic constant (9 x 10^9 N m^2/C^2), q1 and q2 represent the charges of the electron and proton respectively (q1 = -1.6 x 10^-19 C, q2 = +1.6 x 10^-19 C), and r_initial represents the initial distance between the electron and the proton (9 cm = 0.09 m).

The final potential energy (U_final) of the electron-proton system when the electron is 3 cm from the proton is given by:

U_final = k * (q1 * q2) / r_final

where r_final represents the final distance between the electron and the proton (3 cm = 0.03 m).

According to the principle of conservation of energy, the initial potential energy is equal to the sum of the final potential energy and the kinetic energy (K) of the electron:

U_initial = U_final + K

Since the electron is initially at rest, the initial kinetic energy (K_initial) is zero.

Therefore, we have:

U_initial = K_final

Substituting the values into the equations:

k * (q1 * q2) / r_initial = k * (q1 * q2) / r_final + K_final

Simplifying the equation and solving for K_final:

K_final = (k * (q1 * q2) / r_initial) - (k * (q1 * q2) / r_final)

Substituting the values:

K_final = (9 x 10^9 N m^2/C^2) * ((-1.6 x 10^-19 C) * (+1.6 x 10^-19 C)) / 0.09 m - ((9 x 10^9 N m^2/C^2) * ((-1.6 x 10^-19 C) * (+1.6 x 10^-19 C)) / 0.03 m)

Calculating the value of K_final:

K_final = -2.56 x 10^-19 J

The final kinetic energy (K_final) is -2.56 x 10^-19 J.

To find the speed (v) of the electron, we can use the equation:

K_final = (1/2) * m * v^2

where m represents the mass of the electron (9.11 x 10^-31 kg).

Substituting the values and solving for v:

-2.56 x 10^-19 J = (1/2) * (9.11 x 10^-31 kg) * v^2

Simplifying the equation:

v^2 = (-2.56 x 10^-19 J * 2) / (9.11 x 10^-31 kg)

v^2 = -5.62 x 10^11 m^2/s^2

Taking the square root of both sides:

v = +/- 7.50 x 10^5 m/s

Since the electron is moving towards the proton, we take the positive value for the speed:

v ≈ 7.50 x 10^5 m/s

Therefore, the electron will be moving at approximately 7.50 x 10^5 m/s when it is 3 cm from the proton.

To determine the speed of the electron when it is 3 cm from the proton, we can use the principle of conservation of energy. The initial potential energy of the electron is converted into kinetic energy as it moves closer to the proton.

1. First, we need to calculate the initial potential energy of the electron at a distance of 9 cm from the proton. The potential energy between two charged particles can be calculated using the formula:

Potential Energy = (K * q1 * q2) / r

where K is the Coulomb constant (8.99 × 10^9 N⋅m^2/C^2), q1 and q2 are the charges of the particles (charge of the electron is -1.6 × 10^-19 C, charge of the proton is +1.6 × 10^-19 C), and r is the distance between the particles (9 cm = 0.09 m).

Plugging in the values, we get:

Potential Energy = (8.99 × 10^9 N⋅m^2/C^2) * (-1.6 × 10^-19 C) * (1.6 × 10^-19 C) / 0.09 m

Potential Energy = -256 J (negative sign indicates attraction between opposite charges)

2. Next, we can calculate the final potential energy of the electron when it is 3 cm from the proton. Using the same formula, but with r = 0.03 m:

Potential Energy = (8.99 × 10^9 N⋅m^2/C^2) * (-1.6 × 10^-19 C) * (1.6 × 10^-19 C) / 0.03 m

Potential Energy = - 1706.67 J

3. The change in potential energy is given by the difference between the initial and final potentials energies:

ΔPE = Final Potential Energy - Initial Potential Energy

ΔPE = (-1706.67 J) - (-256 J) = -1450.67 J

4. According to the conservation of energy, the change in potential energy is equal to the change in kinetic energy:

ΔKE = ΔPE

ΔKE = -1450.67 J

5. The kinetic energy of the electron can be given by the equation:

Kinetic Energy = (1/2) * m * v^2

where m is the mass of the electron (9.11 × 10^-31 kg) and v is its velocity.

So, we can rewrite the equation as:

(1/2) * m * v^2 = -1450.67 J

6. Rearranging the equation to solve for v, we get:

v = √((-2 * ΔKE) / m)

Plugging in the values, we get:

v = √((-2 * -1450.67 J) / 9.11 × 10^-31 kg)

v = √(2 * 1450.67 J / 9.11 × 10^-31 kg)

v ≈ 1.57 × 10^7 m/s

Therefore, the electron will be moving at approximately 1.57 × 10^7 m/s when it is 3 cm from the proton.

At nine cm, the potential V=kq/.09

at three cm, the potential V=kq/.03

energy gained by electron=differencePotential*e

1/2 mv^2=kq(1/.03-1/.09)e

solve for v