Question

Wonder Woman and Superman fly to an altitude of 1590 km,1590 km, carrying between them a chest full of jewels that they intend to put into orbit around Earth. They want to make this tempting treasure inaccessible to their evil enemies who are trying to gain possession of it, yet keep it available for themselves for future use when they retire and settle down. But perhaps the time to retire is now! They accidentally drop the chest, which leaves their weary hands at rest, and discover that they are no longer capable of catching it as it falls into the Pacific Ocean. At what speed vfvf does the chest impact the surface of the water? Ignore air resistance (in reality, it would make large difference). The radius and mass of Earth are 6370 km6370 km and 5.98×1024 kg,5.98×1024 kg, respectively.

Answer 1

Answer:

Explanation:

Just start with the trivial. If gravity was a constant then E = mgh and 1/2 m v^2 = E

so v=sqrt(2gh)=sqrt(2*9.8*1590000) = 5582m/s

Now this will be too high as gravity reduces with distance.

However it is still true that 1/2mv^2 = loss of gravitational potential energy

so 1/2 v^2 = loss of gravitational potential ( i.e a field without considering mass )

As g = GM/ Ro^2 and P = - GM/R

the Po = - 9.8 * (6370*10^3)= - 62.4 * 10 ^ 6 J/kg

P1= Po * 6370/(6370+1590) = - 49.93 * 10 ^ 6 J/kg

find the CHANGE and then from that the velocity

ie v = sqrt(2*( P1 - Po)) = 5094 m/s

Note how it is a bit smaller than the first estimate but not by such a margin that they are unrecognizably different.

Answer 2

Answer:

5001.51 m/s

Explanation:

Gravitation potential of a body in orbit from the center of the earth is given as

Pg = -GM/R

Where G is the gravitational constant 6.67x10^-11 N-m^2kg^-2

M is the mass of the earth = 5.98x10^24 kg

R is the distance from that point to the center of the earth = r + Re

r is the distance above earth surface, Re is the earth's radius.

R = 1590 km + 6370 km = 7960 km

Pg = -(6.67x10^-11 x 5.98x10^24)/7960x10^3

Pg = -50108793.97 J/kg

The negative sign means that the gravitational potential is higher away from earth than it is at the earth's surface (it shows convention).

This indicates the kinetic energy per kilogram that the chest of jewel will fall with to earth.

Gravitation Potential on earth's surface is

Pg = -GM/Re

= -(6.67x10^-11 x 5.98x10^24)/6370x10^3 = -62616326.53 J/kg

Difference in gravity potential = -50108793.97 - (-62616326.53)

= 12507532.56 J/kg

The velocity V of the jewel chest will be

0.5v^2 = 12507532.56

V^2 = 25015065.12

V = 5001.51 m/s