Nuclear fusion reactions require that positively charged nuclei be brought into close proximity, against the electrostatic repulsion. As a simple example, suppose a proton is fired at a second, stationary proton from a large distance away. What kinetic energy must be given to the moving proton to get it to come within 1.00e10-15 m of the target? Assume that there is a head-on collision and that the target is fixed in place.

KE>=PE

1/2 m v^2=kq1q2/d where q1, q2 are the two charges, k is the coulomb constant, and d is the distance given.

The above ignores relativistic effects.

To calculate the kinetic energy required to bring the moving proton within 1.00e-15 m of the target proton, we need to consider the electrostatic potential energy and the conservation of mechanical energy.

The electrostatic potential energy is given by the equation:

U = k * (q1 * q2) / r

Where:
- k is Coulomb's constant (k = 8.99e9 N m²/C²)
- q1 and q2 are the charges of the two protons (both are the charge of an elementary charge, q = 1.60e-19 C)
- r is the distance between the protons (r = 1.00e-15 m)

Since the target proton is stationary, it will have no kinetic energy. The initial kinetic energy of the moving proton will be completely converted into potential energy at the closest approach distance.

Therefore, the kinetic energy of the moving proton must be equal to the absolute value of the electrostatic potential energy when r = 1.00e-15 m.

K = |U|

Substituting the values into the equation:

K = (8.99e9 N m²/C²) * ((1.60e-19 C)²) / (1.00e-15 m)
K = 2.304e-13 J

Therefore, the kinetic energy of the proton must be 2.304e-13 Joules to bring it within 1.00e-15 meters of the target proton.

To determine the kinetic energy required for the moving proton to come within a certain distance of the target proton, we can use the principle of conservation of mechanical energy.

The initial mechanical energy of the system, consisting of the two protons, is equal to the sum of their kinetic energies. At the final distance of 1.00e-15 m, the system potential energy is zero since they have come close enough for their repulsion to be negligible.

The potential energy is given by the equation:

U = k * (q1 * q2) / r,

where U is the potential energy, k is the electrostatic constant (8.99 x 10^9 N m^2/C^2), q1 and q2 are the charges of the protons (both equal to 1.6 x 10^-19 C), and r is the distance between the protons.

As the initial potential energy is zero, the initial mechanical energy is equal to the initial kinetic energy of the moving proton. We can express this as:

KE_initial = KE1 = 0.5 * m * v^2,

where m is the mass of the proton (1.67 x 10^-27 kg) and v is its initial velocity.

At the final distance, the kinetic energy is given by:

KE_final = KE2 = 0.5 * m * u^2,

where u is the final velocity of the moving proton.

Since mechanical energy is conserved, we can equate the initial and final kinetic energy:

KE_initial = KE_final,

0.5 * m * v^2 = 0.5 * m * u^2.

Simplifying and rearranging the equation, we get:

v^2 = u^2.

Taking the square root of both sides, we have:

v = u.

So, the initial and final velocities are equal. Therefore, we need to find the velocity of the proton when it reaches the final distance of 1.00e-15 m.

To do this, we can use the principle of conservation of linear momentum. Since the system is closed, the initial and final linear momenta are equal. Therefore, we can equate them as:

p_initial = p_final,

m * v = m * u.

Canceling out the mass "m" on both sides, we get:

v = u.

This means that the initial velocity "v" is equal to the final velocity "u." So, we can determine the velocity of the moving proton when it comes within the desired distance.

Now, we can calculate the velocity by using Coulomb's law to find the electric force between the protons. At the final distance of 1.00e-15 m, the electric force between the protons will be equal to the centripetal force required to keep the moving proton in a circular orbit around the stationary proton. The centripetal force is given by:

F_c = m * u^2 / r,

where F_c is the centripetal force and r is the distance between the protons.

The electric force can be calculated using Coulomb's law:

F_e = k * (q1 * q2) / r^2,

where F_e is the electric force, k is the electrostatic constant, q1 and q2 are the charges of the protons, and r is the distance between them.

Since the electric force is equal to the centripetal force, we can equate them:

k * (q1 * q2) / r^2 = m * u^2 / r.

Simplifying and rearranging the equation, we get:

u^2 = (k * (q1 * q2)) / (m * r).

Substituting the known values:

u^2 = (8.99 x 10^9 N m^2/C^2) * (1.6 x 10^-19 C)^2 / (1.67 x 10^-27 kg * 1.00e-15 m).

Simplifying further, we can solve for u:

u = sqrt((8.99 x 10^9 N m^2/C^2) * (1.6 x 10^-19 C)^2 / (1.67 x 10^-27 kg * 1.00e-15 m)).

Evaluating this expression will give us the final velocity "u" of the moving proton.