An electron is released from rest at a distance of 9.00 cm from a proton. If the proton is held in place, how fast will the electron be moving when it is 3.00 cm from the proton? (m_el= 9.11 x 10^-31 kg, e = 1.60 × 10^-19 C, k = 1/4πε₀ = 8.99 × 10⁹ N ∙ m²/C²) A) 75.0 m/s B) 106 m/s C) 130 m/s D) 1.06 × 10³ m/s E) 4.64 × 10⁵ m/s

Question

Quizplus · Accepted Answer

We can use conservation of energy to solve this problem. The initial total energy of the electron-proton system is zero (since both are initially at rest), and the final total energy is given by the sum of the kinetic energies of the electron and proton when the electron is 3.00 cm from the proton. The potential energy at any distance between the electron and proton is given by the Coulomb potential energy equation:

$U = -\dfrac{kQq}{r}$

where k is the Coulomb constant, Q and q are the charges of the particles, and r is the distance between them.

At the initial distance of 9.00 cm, the potential energy of the system is:

$U_i = -\dfrac{(8.99 	imes 10^9 	ext{ N}\cdot	ext{m}^2/	ext{C}^2)(1.60 	imes 10^{-19} 	ext{ C})(1.60 	imes 10^{-19} 	ext{ C})}{0.09 	ext{ m}} = -3.56 	imes 10^{-18} 	ext{ J}$

At a distance of 3.00 cm, the potential energy of the system is:

$U_f = -\dfrac{(8.99 	imes 10^9 	ext{ N}\cdot	ext{m}^2/	ext{C}^2)(1.60 	imes 10^{-19} 	ext{ C})(1.60 	imes 10^{-19} 	ext{ C})}{0.03 	ext{ m}} = -3.21 	imes 10^{-17} 	ext{ J}$

The change in potential energy as the electron moves from 9.00 cm to 3.00 cm is:

$\Delta U = U_f - U_i = (-3.21 	imes 10^{-17} 	ext{ J}) - (-3.56 	imes 10^{-18} 	ext{ J}) = -2.85 	imes 10^{-17} 	ext{ J}$

This change in potential energy is equal to the kinetic energy gained by the electron, since there is no external work done on the system. Therefore,

$\dfrac{1}{2}mv_e^2 = -\Delta U = 2.85 	imes 10^{-17} 	ext{ J}$

Solving for the speed of the electron, we get:

$v_e = \sqrt{\dfrac{2(-\Delta U)}{m_e}} = \sqrt{\dfrac{2(2.85 	imes 10^{-17} 	ext{ J})}{9.11 	imes 10^{-31} 	ext{ kg}}} \approx 106 	ext{ m/s}$

Therefore, the answer is B.

An Electron Is Released from Rest at a Distance of 9.00