A 5.0 104 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 4.5 105 N over a distance of 2600 km. What is the final speed of the probe?

m/s

This is the same thing as performing negaive work on the space proble. Multiple retrorocket force x distance to get the kinetic energy change. Get the final speed from the new lower kinetic energy

To determine the final speed of the space probe, we can use the principle of work and energy.

The work done on an object is given by the product of the force applied on the object and the distance over which the force is applied. In this case, the retrorockets generate a force of 4.5 * 10^5 N and apply it over a distance of 2600 km.

First, let's convert the distance from kilometers to meters:
2600 km * 1000 m/km = 2.6 * 10^6 m

Now, we can calculate the work done on the space probe:
Work = Force * Distance
Work = 4.5 * 10^5 N * 2.6 * 10^6 m

Next, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. In this case, the initial kinetic energy of the space probe is given by its mass and initial speed.

The initial kinetic energy, KE_initial, is given by:
KE_initial = (1/2) * mass * (initial speed)^2
KE_initial = (1/2) * (5.0 * 10^4 kg) * (11000 m/s)^2

Finally, we can solve for the final speed, v_final, using the work-energy principle:
Work = KE_final - KE_initial
KE_final = Work + KE_initial
(1/2) * mass * (v_final)^2 = Work + (1/2) * mass * (initial speed)^2

Rearranging the equation and solving for v_final:
(v_final)^2 = (2 * Work / mass) + (initial speed)^2
v_final = sqrt((2 * Work / mass) + (initial speed)^2)

Plug in the values:
v_final = sqrt((2 * (4.5 * 10^5 N * 2.6 * 10^6 m) / (5.0 * 10^4 kg)) + (11000 m/s)^2)

Calculating the final speed:
v_final ≈ 10864.3 m/s

Therefore, the final speed of the space probe is approximately 10864.3 m/s.