Question

4- Due to his knowledge of physics and the interest in the nature, you've got a summer job with an organization that wants to put a satellite in orbit to monitor the quantity of chlorine ions in the upper atmosphere above North America. It has been determined that the satellite must collect samples at an altitude of 100 miles above the Earth's surface. Unfortunately, by that point the air resistance would make satellite will remain in orbit for a very short time period, which would not be useful. You suggest that an elliptical orbit would be near the ground on North America, where the data are desired and also away from the Earth, the satellite out of almost the totality of the atmosphere, on the other side of the planet. His colleague estimated that the satellite would be traveling to 10,000 mi / h when this farthest away from the Earth at an altitude of 1,000 miles. How quickly traveled the satellite when it took air samples if it does not consider air friction? Use conservation of energy to determine its answer.
Please determine the resultnusing the conservation of energy. Thanks

Answer 1

1000miles=1609.34km

At this height acceleration due to gravity is $g'=g\left(\frac{R_e}{R_e+h} \right )^2=6.26m/s^2$

Total energy at this point is Potential energy+Kinetic energy=$mg'h+0.5mv^2=m(6.26\times1609.34\times10^3+0.5\times4470.4^2)J=2\times10^7mJ$

100miles=160.93km

At this height acceleration due to gravity is $g''=g\left(\frac{R_e}{R_e+h} \right )^2=9.33m/s^2$

Total energy at this point is Potential energy+Kinetic energy=$mg''h+0.5mv^2=m(9.33\times160.93\times10^3+0.5\times v^2)J$

From conservation of energy, total energy at two points should be equal so,

$m(9.33\times160.93\times10^3+0.5\times v'^2)=2\times10^7m$

$\implies v'=$6082.52m/s=13606.21mi/hr