The MAVEN spacecraft, a NASA mission that is studying the Martian atmosphere, launched from Cape Canaveral in November 2013 on an Atlas V rocket. The rocket burns about 280,000 kg of fuel over 4 minutes (you can assume dm/dt is constant over that interval) and ejects the fuel at speeds of 4000 m/s. How much thrust does the rocket provide?
To find the thrust provided by the rocket, we can use the principle of conservation of momentum. The momentum of the rocket before and after the ejection of the fuel must be equal.
The momentum of the rocket is given by the formula: momentum = mass × velocity.
Initially, the mass of the rocket is the sum of its own mass (m) and the mass of the fuel (dm). The velocity of the rocket before the fuel is ejected is 0 since it is at rest.
Therefore, before the fuel is ejected:
Initial momentum = (m + dm) × 0 = 0
After the fuel is ejected, the mass of the rocket decreases by dm, and the rocket moves with a velocity of v (the speed at which the fuel is ejected).
Therefore, after the fuel is ejected:
Final momentum = m × v
According to the principle of conservation of momentum, the initial and final momenta are equal. Thus, we can equate the two expressions and solve for the thrust.
0 = m × v
Rearranging the equation, we find that the amount of thrust (F) provided by the rocket is given by:
F = m × v
Now, let's calculate the thrust provided by the rocket using the given values.
Given:
Mass of fuel burned (dm) = 280,000 kg
Ejection velocity (v) = 4000 m/s
The mass of the rocket (m) is not given in the question, but we know that dm/dt (mass flow rate) is constant over the 4-minute interval. Since dm/dt is constant, we can assume it represents the rate at which the fuel is burned. Therefore, we can calculate m by multiplying the mass flow rate (dm/dt) by the time interval (4 minutes = 240 seconds).
Mass flow rate (dm/dt) = dm / dt
Since dm/dt is constant, we have:
dm / dt = (mass of fuel burned) / (time interval) = 280,000 kg / 240 s
Substituting this value into the equation, we can find m:
m = (dm / dt) × (time interval) = (280,000 kg / 240 s) × (240 s)
Now we have the value of m, which we can use to calculate the thrust:
F = m × v
Substituting the values, we can find the thrust provided by the rocket.
To determine the thrust provided by the rocket, we can use Newton's second law of motion, which states that force (thrust) is equal to the rate of change of momentum.
The equation is as follows:
F = dp/dt
Where:
F is the thrust force
dp is the change in momentum
dt is the change in time
In this case, the momentum change (dp) is equal to the mass of fuel burned (dm) times the velocity of fuel ejected (v). Therefore, we can write it as:
dp = dm * v
Since dm/dt is assumed to be constant over the given interval (4 minutes), we can rewrite it as:
dm = dm/dt * dt
So now, dp becomes:
dp = (dm/dt * dt) * v
Substituting the values given:
dp = (280,000 kg / (4 minutes * 60 seconds)) * (4000 m/s)
Now, we need to convert the 4 minutes to seconds:
dm = (280,000 kg / (240 s)) * (4000 m/s)
Simplifying the equation:
dm = 280,000 kg / 240 s * 4000 m/s
dm = 4,666.67 kg * 4000 m/s
Now, we can calculate the momentum change, dp:
dp = 18,666,666.7 kg * m/s
Finally, we can substitute dp into the thrust equation:
F = dp/dt = 18,666,666.7 kg * m/s / 240 s
Calculating this expression, the thrust provided by the rocket is approximately:
F ≈ 77,777.78 N
Therefore, the rocket provides a thrust of approximately 77,777.78 Newtons.