A 4.5 104-kg space probe is traveling at a speed of 15000 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 3.0 105 N over a distance of 2400 km. What is the final speed of the probe?
the retrorockets do work (force*distance).
Intial KE-FinalKE= rocket work
calculate final KE, then velocity.
To determine the final speed of the space probe, we can use the principles of Newton's laws of motion and the work-energy theorem.
First, we need to convert the given force, distance, and mass into appropriate units.
Given:
Mass of the probe (m) = 4.5 * 10^4 kg
Initial speed of the probe (u) = 15000 m/s
Force exerted by retrorockets (F) = 3.0 * 10^5 N
Distance over which the force is applied (s) = 2400 km = 2.4 * 10^6 m
Now let's use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy.
The work done by the retrorockets can be calculated using the formula:
Work (W) = Force (F) * Distance (s)
W = (3.0 * 10^5 N) * (2.4 * 10^6 m)
W = 7.2 * 10^11 J
Since the initial kinetic energy of the probe is given as:
Initial kinetic energy (KE_initial) = 0.5 * mass (m) * (initial speed (u))^2
KE_initial = 0.5 * (4.5 * 10^4 kg) * (15000 m/s)^2
KE_initial = 1.0125 * 10^13 J
The final kinetic energy (KE_final) can be calculated as:
KE_final = KE_initial - W
KE_final = 1.0125 * 10^13 J - 7.2 * 10^11 J
KE_final = 9.925 * 10^12 J
To get the final speed (v) of the probe, we can use the formula for kinetic energy:
Final kinetic energy (KE_final) = 0.5 * mass (m) * (final speed (v))^2
Plugging in the values, we have:
9.925 * 10^12 J = 0.5 * (4.5 * 10^4 kg) * (v)^2
Simplifying the equation, we get:
(v)^2 = (2 * 9.925 * 10^12 J) / (4.5 * 10^4 kg)
(v)^2 = 44.0556 * 10^8 J/kg
Taking the square root of both sides, we find:
v = sqrt(44.0556 * 10^8 J/kg)
v ≈ 2.1 * 10^4 m/s
Therefore, the final speed of the space probe after firing the retrorockets is approximately 2.1 * 10^4 m/s.