The space shuttle, with an initial mass M = 2.41E+6 kg, is launched from the surface of the earth with an initial net acceleration a = 26.1 m/s2. The rate of fuel consumption is R = 6.90E+3 kg/s. The shuttle reaches outer space with a velocity of vo = 4632 m/s, and a mass of Mo = 1.45E+6 kg. How much fuel must be burned after this time to reach a velocity vf = 4821 m/s?
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To determine how much fuel must be burned to reach a certain velocity, we need to use the principle of conservation of momentum.
The momentum of the shuttle before it burns any fuel is given by:
Initial momentum (p_initial) = mass (M) * initial velocity (v_initial)
After burning some fuel and reaching a new velocity, the momentum will be:
Final momentum (p_final) = (mass after burning fuel (M_fuel)) * final velocity (v_final)
The change in momentum is equal to the momentum lost due to burning fuel:
Change in momentum (Δp) = p_initial - p_final
We can now calculate the amount of fuel burned by dividing the change in momentum by the exhaust velocity of the fuel (V_e). This is given by the formula:
Amount of fuel burned (Δm_fuel) = Δp / V_e
First, let's calculate the initial momentum:
p_initial = M * vo
Next, let's calculate the final momentum:
p_final = Mo * vf
Then, calculate the change in momentum:
Δp = p_initial - p_final
Now, we need to determine the exhaust velocity, which is given by the difference between the initial velocity and the net acceleration multiplied by the burn time:
V_e = vo - a * t
To find the burn time (t), we need to consider that the rate of fuel consumption (R) is equal to the change in mass (Δm) divided by the burn time (t), so:
R = Δm / t
Rearranging the equation, we can find the change in mass (Δm):
Δm = R * t
Substituting this value into the equation for exhaust velocity gives:
V_e = vo - a * (Δm / R)
Now, we can plug in the values and solve for the amount of fuel burned:
Δm_fuel = Δp / V_e
Once we have Δm_fuel, we can subtract it from the initial mass (M) to get the amount of fuel that must be burned after reaching the desired velocity.
Finally, let's put all the values into the equations and calculate the result.