An electron of mass 9.1E-31 kg is released from rest at a distance of 7.1 E-10 m from a fixed electron. The force on the movable electron is given by the function F = 2.3E-28/r^2, where r is the distance between the electrons. Find the maximum velocity (in meters/second) of the moving electron after release.
^ we're supposed to use mechanical energy theorums to solve this problem
I assume you are a calculus person.
work done on electron= final KE of electron
and from final KE, you calaculate the velocity. What is work done on the electron?
work= intgeral force dot dx integrated from original distance to final distance.
Now the final distance here is not specifed, and velocity will depend on that. If you are unsure, let the final distance be rfinal.
If you use zero as the final distance between the electrons, you complicate the integral as you will see.
You can also use potential theory here, I am not so certain you have had that yet.
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To solve this problem using the mechanical energy theorem, we need to analyze the energy changes of the system.
Initially, when the electron is at rest, it has potential energy due to the electrostatic force between the electrons, given by U_initial = -2.3E-28 / r_initial.
When the electron moves to a distance r from the fixed electron, its potential energy changes to U_final = -2.3E-28 / r.
According to the conservation of mechanical energy, the total mechanical energy of the system remains constant. Thus, the initial mechanical energy (considering only potential energy) is equal to the final mechanical energy (considering only kinetic energy).
Assuming the initial kinetic energy of the electron is zero, we can set the total mechanical energy equation as follows:
U_initial = U_final + K_final
Since the initial kinetic energy is zero, K_final can be written as:
K_final = U_initial - U_final
Substituting the values:
0 = (-2.3E-28 / r_initial) - (-2.3E-28 / r) + K_final
Simplifying:
2.3E-28 / r_initial = 2.3E-28 / r + K_final
Now, we can calculate the maximum velocity (v_max) of the moving electron by using the formula for kinetic energy:
K_final = (1/2) * m * v_max^2
Substituting the given values:
2.3E-28 / r_initial = 2.3E-28 / r + (1/2) * (9.1E-31) * v_max^2
Dividing the equation through by 2.3E-28:
1 / r_initial = 1 / r + (v_max^2 / (2 * 9.1E-31))
Simplifying further:
(v_max^2 / (2 * 9.1E-31)) = (1 / r_initial) - (1 / r)
Now, we can solve for v_max:
v_max^2 = (2 * 9.1E-31) * ((1 / r_initial) - (1 / r))
Taking the square root of both sides:
v_max = sqrt((2 * 9.1E-31) * ((1 / r_initial) - (1 / r)))
Finally, substitute the given values for r_initial and r, and calculate v_max using the formula above.