Final answer:
The speeds of the two protons when they are 8.0 x 10^-15 m apart would be approximately 3258.6 m/s.
Step-by-step explanation:
To find the speeds of two protons when they are 8.0 x 10^-15 m apart, we can use the principle of conservation of energy. At their initial position, the protons have potential energy due to their separation. As they move closer together, this potential energy is converted into kinetic energy. We can set up the following equation:
Initial potential energy = Final kinetic energy
PE_initial = KE_final
Using the formula for potential energy and the equation for kinetic energy, we have:
(kQ^2 / r_initial) = (1/2)m(v_final^2)
Where k is the electrostatic constant, Q is the charge of each proton, r_initial is the initial separation, m is the mass of each proton, and v_final is the final speed.
Plugging in the given values, we get:
(9 x 10^9 Nm^2/C^2)(1.6 x 10^-19 C)^2 / (10^-15 m) = (1/2)(1.67 x 10^-27 kg)(v_final^2)
Simplifying, we can solve for v_final:
v_final^2 = ((9 x 10^9 Nm^2/C^2)(1.6 x 10^-19 C)^2 / (10^-15 m)) / (1.67 x 10^-27 kg)
v_final ≈ 3258.6 m/s
Therefore, the speeds of the two protons when they are 8.0 x 10^-15 m apart would be approximately 3258.6 m/s.