A 6090kg space probe mobing nose-first toward Jupiter at 105m/s relative to the Sun, fires its rocket engine ejecting 80 kg of exhaust at a speed of 253 m/s relative to the space probe. What is the final velocity of the probe?
I should have gotten 108m/s Can you show me the correct way to solve this problem? Thanks so much for your time and effort. I really would apprecitate it!
Momentum:
80(253)=(6090-80)delta v
deltav=?
This is an approximation. A better solution is to take the average mass of the spacecraft as (6090-80/4), but it wont matter much.
There is an exact method in Calculus, but I assume you are not a calculus person.
To solve this problem, we can start by applying the principle of conservation of momentum. The total momentum before the rocket engine fires is equal to the total momentum after the rocket engine fires.
Before the engine fires, the momentum of the space probe is given by:
P_initial_probe = mass_probe * V_initial_probe
where mass_probe is the mass of the space probe and V_initial_probe is its initial velocity relative to the Sun.
After the engine fires, the momentum of the space probe is given by:
P_final_probe = (mass_probe - mass_exhaust) * V_final_probe
where mass_exhaust is the mass of the exhaust and V_final_probe is the final velocity of the space probe relative to the Sun.
The momentum of the exhaust is given by:
P_exhaust = mass_exhaust * V_exhaust
where V_exhaust is the velocity of the exhaust relative to the space probe.
Since the total momentum is conserved, we have:
P_initial_probe = P_final_probe + P_exhaust
Substituting the momentum equations, we get:
mass_probe * V_initial_probe = (mass_probe - mass_exhaust) * V_final_probe + mass_exhaust * V_exhaust
Given the values:
mass_probe = 6090 kg
V_initial_probe = 105 m/s
mass_exhaust = 80 kg
V_exhaust = 253 m/s
we can solve for V_final_probe:
6090 kg * 105 m/s = (6090 kg - 80 kg) * V_final_probe + 80 kg * 253 m/s
634,950 kg * m/s = 6,010,370 kg * V_final_probe + 20,240 kg * m/s
Subtracting 20,240 kg * m/s from both sides and rearranging the equation, we get:
634,950 kg * m/s - 20,240 kg * m/s = 6,010,370 kg * V_final_probe
614,710 kg * m/s = 6,010,370 kg * V_final_probe
Now, divide both sides by 6,010,370 kg to solve for V_final_probe:
V_final_probe = 614,710 kg * m/s / 6,010,370 kg
V_final_probe ≈ 102.25 m/s
However, it seems like you calculated an answer of 108 m/s. Please double-check your calculations to make sure there are no errors.
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the rocket engine is fired is equal to the total momentum after the rocket engine is fired.
The momentum of the space probe before the engine is fired is given by:
Initial momentum = mass of the space probe * velocity of the space probe
Initial momentum = 6090 kg * 105 m/s = 639,450 kg*m/s
The momentum of the exhaust gases after being ejected from the space probe is given by:
Momentum of exhaust gases = mass of exhaust * velocity of exhaust
Momentum of exhaust gases = 80 kg * 253 m/s = 20,240 kg*m/s
Let the final velocity of the probe be v_f.
According to the principle of conservation of momentum:
Initial momentum = Final momentum
639450 kg*m/s = (6090 kg - 80 kg) * v_f
Simplifying the equation:
639450 kg*m/s = 6010 kg * v_f
v_f = 639450 kg*m/s / 6010 kg
v_f ≈ 106.57 m/s
So, the final velocity of the probe is approximately 106.57 m/s, which is close to your calculated value of 108 m/s.
Conservation of momentum:
6090(105) = 80(-253) + 6010(Vf)
Vf = 109.77 m/s