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.
To find the maximum velocity of the moving electron, we need to analyze the conservation of mechanical energy. The formula for mechanical energy is E = K + U, where E is the mechanical energy, K is the kinetic energy, and U is the potential energy.
Initially, when the electron is at rest, the only energy present is the potential energy given by U = -k/r, where k is the electrostatic constant (8.99E9 Nm^2/C^2) and r is the initial distance between the electrons (7.1E-10 m).
When the electron is released, it converts the potential energy into kinetic energy as it moves towards the fixed electron. At maximum velocity, all the potential energy is converted into kinetic energy, so we can equate U to zero and solve for K.
U = 0
-k/r = 0
Solving for r, we have:
k/r = 0
8.99E9 / 7.1E-10 = 0
Now, let's calculate the maximum velocity using the equation for kinetic energy:
K = 1/2 * m * v^2
Where m is the mass of the electron (9.1E-31 kg) and v is the maximum velocity we are trying to find.
Setting the kinetic energy equal to the initial potential energy:
K = -U
1/2 * m * v^2 = k / r
1/2 * 9.1E-31 * v^2 = 8.99E9 / 7.1E-10
Solving for v^2:
v^2 = (8.99E9 * 2) / (9.1E-31 * 7.1E-10)
v^2 = 16.492
Finally, taking the square root of both sides to find the maximum velocity:
v = sqrt(16.492)
v ≈ 4.06 x 10^0 m/s
Therefore, the maximum velocity of the moving electron after release is approximately 4.06 m/s.