Four point charges of 89.67, 16.46, 12.1, and 19.76 microcoulombs are located at the corners of a rectangle of length 42 cm and width of 638 cm, as shown.

A sodium ion is placed at the point halfway between 16.46 and 12.1, and then released from rest. What will be its speed after it has traveled a distance of 45 km from where it was released? Assume it didn't bump into anything along the way, including any atoms or other charged particles, and ignore any effects that could be attributed to earth's gravity or magnetic field. In other words, keep the doggone problem simple! I've given you enough to think about just to be able to figure out the mass of a sodium ion!

To find the final speed of the sodium ion after it has traveled a distance of 45 km, we need to calculate the total work done on the ion.

The work done can be calculated using the equation: Work = Change in Kinetic Energy (ΔKE)

Since the sodium ion starts from rest, its initial kinetic energy is zero (KE_initial = 0). Therefore, the work done is equal to the final kinetic energy (KE_final).

To calculate the work done, we need to find the electric potential energy (PE) at the starting point and the final point. The change in potential energy (ΔPE) will be equal to the work done (W).

The electric potential energy of a point charge is given by the equation:
PE = k * q1 * q2 / r

Where:
- k = Coulomb's constant (8.99 x 10^9 N m^2/C^2)
- q1 and q2 are the magnitudes of the charges (in Coulombs)
- r is the distance between the charges (in meters)

Let's calculate the electric potential energies at the starting point and the final point.

1) Starting Point:
The starting point is halfway between 16.46 µC and 12.1 µC charges. The charges at the corners form a rectangle of length 42 cm (0.42 m) and width 638 cm (6.38 m).

The distance between the sodium ion and the 16.46 µC charge is half of the length of the rectangle, i.e. 0.21 m.
PE_start = k * q_sodium * q_16.46 / r_start

Similarly, the distance between the sodium ion and the 12.1 µC charge is also 0.21 m.
PE_start = k * q_sodium * q_12.1 / r_start

2) Final Point:
The final point is 45 km away from the starting point. Converting 45 km to meters: 45,000 m.

PE_final = k * q_sodium * q_total / r_final

Here, q_total is the sum of the magnitudes of all the charges:
q_total = |q_89.67| + |q_16.46| + |q_12.1| + |q_19.76|

Next, calculate the difference in potential energy (ΔPE = PE_final - PE_start) and equate it to the work done (W).

W = ΔPE = PE_final - PE_start

Now, equate the work done to the change in kinetic energy:
W = ΔKE

Finally, solve for the final kinetic energy (KE_final) and use it to calculate the final speed of the sodium ion.

KE_final = W

v_final = √(2 * KE_final / m_sodium)

Where:
- KE_final is the final kinetic energy
- m_sodium is the mass of the sodium ion

Note: The mass of a sodium ion is approximately 23 atomic mass units (amu) or 3.82 x 10^-26 kg.