A 7.00 104-kg space probe is traveling at a speed of 12000 m/s through deep space. Retrorockets are fired along the line of motion to reduce the probe's speed. The retrorockets generate a force of 4.00 105 N over a distance of 2700 km. What is the final speed of the probe?

To find the final speed of the probe, we can use the concept of work and energy. The work done on an object is equal to the change in its kinetic energy.

First, let's convert the given distance from kilometers to meters:
Distance = 2700 km = 2700 × 1000 m = 2,700,000 m

Next, we need to find the work done by the retrorockets. The work done is equal to the force applied multiplied by the displacement. Therefore, we can calculate the work done:
Work = Force × Distance = (4.00 × 10^5 N) × (2,700,000 m) = 1.08 × 10^12 N·m = 1.08 × 10^12 J

Now, let's calculate the initial kinetic energy of the probe. The kinetic energy is given by the equation:
Kinetic Energy = (1/2) × mass × velocity^2

Mass of the probe = 7.00 × 10^4 kg
Initial velocity of the probe = 12000 m/s

Kinetic Energy = (1/2) × (7.00 × 10^4 kg) × (12000 m/s)^2
Kinetic Energy = 5.04 × 10^12 J

Since the work done by the retrorockets is equal to the change in kinetic energy, we can set up the equation:
Work = Final Kinetic Energy - Initial Kinetic Energy

Rearranging the equation, we can solve for the final kinetic energy:
Final Kinetic Energy = Work + Initial Kinetic Energy
Final Kinetic Energy = 1.08 × 10^12 J + 5.04 × 10^12 J
Final Kinetic Energy = 6.12 × 10^12 J

Finally, we can calculate the final velocity of the probe using the final kinetic energy:
Final velocity = √((2 × Final Kinetic Energy) / mass)

Final velocity = √((2 × 6.12 × 10^12 J) / (7.00 × 10^4 kg))
Final velocity ≈ 5596.28 m/s

Therefore, the final speed of the probe is approximately 5596.28 m/s.