Q.45 - Excercises And ProblemsExpert-verified
An astronaut circling the earth at an altitude of is horrified to discover that a cloud of space debris is moving in the exact same orbit as his spacecraft, but in the opposite direction. The astronaut detects the debris when it is away. How much time does he have to fire his rockets and change orbits?
The time required to fire his rockets and change orbits is
height of the astronaut
Radius of earth
distance between debris and astronaut
One of the essential concepts to solve this problem is the utilization of the equations of centripetal and gravitational force.
From them it will be possible to find the speed of the body with which the estimated time can be calculated through the kinematic equations of motion. At the same time for the calculation of this speed it is necessary to clarify that this will remain twice the ship, because as we know by relativity, when moving in the same magnitude but in the opposite direction, with respect to the ship the debris will be double speed.
By equilibrium the centrifugal force and the gravitational force are equal therefore
is mass spacecraft
is Gravitational Universal Constant
is Mass of earth
Radius of earth and orbit
Re-arrange to find the velocity
Replacing with our values we have
From the cinematic equations of motion we have to
Remember that the speed is double for the counter-direction of the trajectories.
Therefore the time required is
In 2014, the European Space Agency placed a satellite in orbit around comet 67P/Churyumov-Gerasimenko and then landed a probe on the surface. The actual orbit was elliptical, but we’ll approximate it as a 50-km-diameter circular orbit with a period of 11 days.a. What was the satellite’s orbital speed around the comet? (Both the comet and the satellite are orbiting the sun at a much higher speed.)
b. What is the mass of the comet?
c. The lander was pushed from the satellite, toward the comet, at a speed of 70 cm/s, and it then fell—taking about 7 hours—to the surface. What was its landing speed? The comet’s shape isirregular, but on average it has a diameter of 3.6 km.
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