Open-Response Homework Problem 24.4 [5 pts] A new summer disaster movie tells th

Question

Open-Response Homework Problem 24.4 [5 pts] A new summer disaster movie tells the story of an astronomer who notices an asteroid passing through a point in space, 19 Earth radii away from the surface of the Earth, with a velocity 100 m/s perpendicular to a line from the asteroid to the center of the earth. The astronomer figures that the asteroid has been captured by the Earth’s gravitational field, and so it is now on an elliptic orbit around our planet. In order to see if the asteroid poses a real threat to the planet, the astronomer does a quick calculation. What is the distance of closest approach for the asteroid, and what is its velocity when it is at that distance? Is it time to panic?

Explanation / Answer

the asteroid will have velocity perpendicular to line joining the centre of the earth and asteroid two times one is shortest distance and the other is longest distance (similar as aphelian and perihelian distances)

here no external torque is acting on the system hence law of conservation of angular momentum is satisfied so Lbefore=Lafter   ie mvlongestrlongest =mvshortestrshortest

(100)(19Re+Re) =vshortrshort

rlong=a(1+e)=20Re rshort=a(1-e) here a=semimajor axis length hence the shortest distance and minimum velocity would be 200m/s and 10Re   because the product of velocity and distance from the centre is same .if the distance from surface will be 9Re

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