A 5.50 104 kg space probe is traveling at a speed of 11000 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 2600 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 by the retrorockets can be calculated using the formula:
Work = Force * Distance * Cosine(θ)

In this case, the force generated by the retrorockets is 4.00x10^5 N and the distance over which the force acts is 2600 km (or 2.6x10^6 m). Since the force is acting along the line of motion, the angle θ between the force and motion is 0 degrees, so the cosine of θ is 1.
Plugging in the values in the equation, we get:
Work = 4.00x10^5 N * 2.6x10^6 m * 1

Next, we can use the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy. Since the probe is initially moving at a speed of 11000 m/s and we want to find the final speed, we can set up the equation:
Work = (1/2) * Mass * (Final Speed^2 - Initial Speed^2)

Rearranging the equation to solve for the final speed:
Final Speed^2 = (2 * Work / Mass) + Initial Speed^2

Now, we can substitute the values we have:
Mass = 5.5x10^4 kg
Initial Speed = 11000 m/s (given)
Work = 4.00x10^5 N * 2.6x10^6 m * 1 (calculated earlier)

After substituting the values and solving for the final speed, we can take the square root to find the final speed in m/s.