An electron is released from a distance of 2 cm from a proton. How fast will the electron be going when it is .5 cm from the proton?
It says to use the work energy thereom, but I don't understand how to use it. any help would be greatly appreciated!
What is the PE of the electron at 2cm? That PE must convert to KE.
To solve this problem using the work-energy theorem, you need to understand the concept of electric potential energy and its relationship to kinetic energy.
The electric potential energy between two charged particles is given by the equation:
PE = -k * (q1 * q2) / r
where PE is the electric potential energy, k is Coulomb's constant (approximately 8.99 × 10^9 Nm²/C²), q1 and q2 are the charges of the particles, and r is the distance between them.
Initially, when the electron is released from a distance of 2 cm from the proton, the potential energy (PE) is given by:
PE_initial = -k * (e * e) / r_initial
where e is the elementary charge (approximately 1.6 × 10^-19 C) and r_initial is the initial distance of 2 cm (or 0.02 m).
When the electron moves to a distance of 0.5 cm from the proton, the potential energy (PE_final) is given by:
PE_final = -k * (e * e) / r_final
where r_final is the final distance of 0.5 cm (or 0.005 m).
According to the work-energy theorem, the change in potential energy (ΔPE) is equal to the work done on the electron, which is also equal to its change in kinetic energy (ΔKE). Mathematically, this is expressed as:
ΔPE = ΔKE
Substituting the equations for PE_initial and PE_final, we have:
- k * (e * e) / r_initial - (- k * (e * e) / r_final) = KE_final - KE_initial
Simplifying the equation, we get:
- k * (e * e) * (1/r_initial - 1/r_final) = KE_final - KE_initial
Since the initial kinetic energy (KE_initial) is zero (as the electron is released from rest), the equation becomes:
- k * (e * e) * (1/r_initial - 1/r_final) = KE_final
Rearranging the equation to solve for KE_final, we have:
KE_final = - k * (e * e) * (1/r_initial - 1/r_final)
Now, to find the speed of the electron at a distance of 0.5 cm, we need to equate the final kinetic energy (KE_final) to (1/2) * m * v^2, where m is the mass of the electron and v is its final velocity.
Since the electron's mass (m) is approximately 9.1 × 10^-31 kg, we can rewrite the equation as:
(1/2) * m * v^2 = - k * (e * e) * (1/r_initial - 1/r_final)
Rearranging to solve for v, we get:
v = sqrt((-2 * k * (e * e) * (1/r_initial - 1/r_final)) / m)
By substituting the known values (constants and distances) into the equation and performing the calculations, you can find the final velocity (v) of the electron.
Remember to convert the distances from centimeters to meters and be mindful of the units used in the equation.