Question

Two protons at rest and separated by \(1.00 \mathrm{~mm}\) are released simultaneously. What is the speed of either at the instant when the two are \(10.0 \mathrm{~mm}\) apart?

Short answer

Answer: The speed of each proton when they are 10 mm apart is approximately \(2.33 \times 10^5\ \text{m/s}\).

Step-by-step solution

Identify the relevant quantities and constants

In this problem, the following quantities and constants are relevant: - Initial separation, \(r_i = 1.00 \text{ mm}\) - Final separation, \(r_f = 10.0 \text{ mm}\) - Charge of the proton, \(q = 1.60 \times 10^{-19} \text{ C}\) - Vacuum permittivity, \(ε_0 = 8.85\times10^{-12} \text{ C}^2/(\text{N m}^2)\) - Mass of a proton, \(m = 1.67 \times 10^{-27} \text{ kg}\)

Calculate the initial potential energy

Based on Coulomb's Law, the initial potential energy, \(U_i\), is given by the formula: $$U_i=\frac{k \cdot q^2}{r_i}=\frac{1}{4πε_0} \cdot \frac{q^2}{r_i}$$ Where \(k = \frac{1}{4πε_0} ≈ 8.99 \times 10^9 \text{ N m}^2/\text{C}^2\) is the electrostatic constant. Plug the known values and calculate \(U_i\): $$U_i=\frac{8.99 \times 10^9\ \text{N}\cdot\text{m}^2/\text{C}^2 \cdot (1.60 \times 10^{-19}\ \text{C})^2}{1.00 \times 10^{-3}\text{ m}}≈2.30 \times 10^{-15}\text{ J}$$

Apply conservation of energy

Since the protons are initially at rest, their total initial energy is equal to their initial potential energy, \(U_i\). When they are 10 mm apart, their total energy will be the sum of the final kinetic energy (\(K_f\)) of each proton and their final potential energy (\(U_f\)). By applying the conservation of energy, we can write: $$U_i = K_f + U_f$$

Calculate the final potential energy

To determine the final potential energy, \(U_f\), use the same formula as in step 2, but replace the initial separation with the final separation: $$U_f=\frac{k \cdot q^2}{r_f}=\frac{1}{4πε_0} \cdot \frac{q^2}{r_f}$$ Plug the known values and calculate \(U_f\): $$U_f=\frac{8.99 \times 10^9\ \text{N}\cdot\text{m}^2/\text{C}^2 \cdot (1.60 \times 10^{-19}\ \text{C})^2}{10.0 \times 10^{-3}\ \text{m}} ≈ 2.30 \times 10^{-16}\ \text{J}$$

Calculate the final kinetic energy

Using the conservation of energy equation from step 3, solve for the final kinetic energy, \(K_f\): $$K_f = U_i - U_f =2.30 \times 10^{-15}\ \text{J}-2.30 \times 10^{-16}\ \text{J} ≈ 2.07 \times 10^{-15}\ \text{J}$$

Determine the speed of the proton

Since both protons have the same mass and final speed, we can divide the final kinetic energy by 2 to find the kinetic energy of either proton, then use the kinetic energy formula \(K_f=\frac{1}{2}mv^2\) to find the speed of either proton (\(v\)). First, find the kinetic energy per proton: $$K_{f, per \ proton}= \frac{K_f}{2} =\frac{2.07 \times 10^{-15}\ \text{J}}{2} ≈ 1.03 \times 10^{-15}\ \text{J}$$ Now, find the speed \(v\) by solving for v in the equation \(K_{f, per \ proton}=\frac{1}{2}mv^2\): $$v=\sqrt{\frac{2\cdot K_{f, per \ proton}}{m}}$$ Substitute the known values and calculate the speed: $$v=\sqrt{\frac{2\cdot 1.03 \times 10^{-15}\ \text{J}}{1.67 \times 10^{-27}\ \text{kg}}} ≈ 2.33 \times 10^{5}\ \text{m/s}$$ So, the speed of either proton when they are 10 mm apart is approximately \(2.33 \times 10^5\ \text{m/s}\).