over nautis aboard the ill-fated Apollo 13 mission achieved the record of being
ID: 1791889 • Letter: O
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over nautis aboard the ill-fated Apollo 13 mission achieved the record of being just a mass of M, which is 6 x 10., The mass of the Moon t M") is 81 times farthest humw of Jim Lovell. Fred Haise, and Jack Swigert became the m from Earth, which is about 100 times the radius of the Farth R than M.. The crew humans Lovell, from Earth at 7:21 pm CT on April 15, 1970 as they passed behind the The 400,000 km or 100R, distance is now defined by NASA to be the beginning space. In the near future, a four passenger Orion spacecraft with a mass of m hed from Cape Canaveral, Florida at an initial speed vo Right now, we are will be launc makin ing no assumption that to is less than, equal to, or greater than the escape speed (a) Using conservation of energy, derive an expression for the speed of the Orion spacecraft v at some distance from the center of the Earth r. Your answer should be in terms of v ve. R, and r (8 points) ,Vex, e (b) Assume that a test launch of a dumgpy Orion spacecraft regulted in an initial speed of 2. How high above the Earth did this dummy spaéecrat go before comi to a stop and returning to the ground e Sc Orion mission is set to Assume that the first astronaut occupigd Apollo 13 flight path to the Moon (hopefully without the equipment failures). (c) the is Orion flight leaves Earthat var·What is the speed of the spacecrat when it gets to its maximum distance from t th (edge of deep spabe) (7 points)?Explanation / Answer
a) From the conservation of energy principle K+U =constant, where K is the kinetic energy of the Orion spacecraft and V is its potential energy.
When it is on the earth, K=1/2mv02 and U=-GMm/Re where v0 is its initial speed and m is its mass, M is the mass of the earth and Re is its radius.
At a distance r from the center of the earth,K=1/2mv2 and U=-GMm/r.
Therefore we have, 1/2mv02-GMm/Re=1/2mv2-GMm/r.
We know that GMm/Re=1/2mvesc2 where vesc is the escape velocity of the spacecraft.
Thus we have , 1/2mv2=1/2mv02 -1/2mvesc2 +GMm/r
i.e.v2=v02-vesc2+2GM/r
or v=(v02-vesc2+2GM/r)1/2
b) With the initial speed of vesc/2 , let the dummy spacecraft reach a distance y above the earth.
Then from the conservation of energy principle, K+U=constant. i.e.0+mgy=1/2m(vesc/2)2
i.e.y=(vesc/2)2/g=(11.2/2)2*106/9.8=3200 km.
c) The new Orion mission reaches a distance of 100Re from the center of the earth.
Again from the conservation of energy principle,1/2mvesc2 -GMm/Re=-GMm/100Re +1/2mv2 where v is the speed of the spacecraft at this distance and also neglecting the Moon's gravitational influence.
Thus v2=vesc2 -99GM/100Re. Substituting the values of vesc=11.2 km/s, G=6.67*10-11Nm2/kg2,M=6*1024 kg,100Re=400000 km, we obtain v=5.2 km/s.
d)The gravitational p.e between the earth and the spacecraft=UES=-GMem/x where x is the distance between the spacecraft and earth.
The gravitational p.e between the moon and the spacecraft=UMS=-GMmm/(100Re-x).
We have UES=UMS.
i.e. -GMEm/x=-GMmm/(100Re-x)
On solving this equation for x we obtain x=3.95*105 km.
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