Question 1158791: A spacecraft can attain a stable orbit 300 kilometers above Earth if it reaches a velocity of 7.7 kilometers per second. The formula for a rocket's maximum velocity v in kilometers per second is v=-0.0098t+c ln R, where t is the firing time in seconds, c is the velocity of the exhaust in kilometers per second, and R is the ratio of the mass of the rocket filled with fuel to the mass of the rocket without fuel. Find the velocity of a spacecraft whose booster rocket has a mass ratio of 24, an exhaust velocity of 2.8 km/s, and a firing time of 20 s. Can the spacecraft achieve a stable orbit 300 km above Earth?
Answer by ikleyn(53419)   (Show Source):
You can put this solution on YOUR website! .
A spacecraft can attain a stable orbit 300 kilometers above Earth if it reaches a velocity of 7.7 kilometers per second.
The formula for a rocket's maximum velocity v in kilometers per second is v=-0.0098t+c ln R, where t is the firing time
in seconds, c is the velocity of the exhaust in kilometers per second, and R is the ratio of the mass of the rocket
filled with fuel to the mass of the rocket without fuel. Find the velocity of a spacecraft whose booster rocket
has a mass ratio of 24, an exhaust velocity of 2.8 km/s, and a firing time of 20 s.
Can the spacecraft achieve a stable orbit 300 km above Earth?
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To solve the problem, we substitute the given numbers t = 20 seconds, c = 2.8 km/s, R = 24 into the given formula
and calculate
v = -0.0098*20 + 2.8*ln(24) = 8.702550725 km/s for the rocket speed.
It is greater than 7.7 kilometers per second, so, according to the context, the spacecraft can achieve a stable orbit.
Solved.
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