Solution to Problem #66 - How Far and How Fast?

Problem #66 Elliptical Orbit

[amanlrwtfsm]

*Sir your omelette is on your shirt* 🔥🤔🔥

[koenth]

Yay I learned something from this problem!! Thank you professor

About the symmetry: I think the centre of the planet is always at one of the foci of an orbital ellipse, rather than at its centre, so in general I do not believe that we have the symmetry that both V's are the same magnitude and opposite directions.
Only a circular orbit, where the foci coincide will have that symmetry.

But of course when it is shot at 90° it starts at perigee if v^2 > MG/R (or at apogee, if v^2 < MG/R, but in that case it falls to the ground immediately).

[koenth]

I chose to set up the quadratic equation for r. It might help others who took this route:

Once in orbit, Energy E = 0.5 mv^2 - mMG/r and Angular momentum L=mvr sin  α are conserved quantities, and hence are 2E/m = v^2 - 2MG/r and L/m = vr sin α

At perigee and apogee, sin α = 1 so at those points v=L/(mr) so that 
0 = -(2E/m) -2MG/r + ( L/(mr) )^2  multiply by r^2
-(2E/m)r^2 -2MGr + (L/m)^2 = ar^2+br+c = 0
with
a= -2E/m = 2MG/R-v0^2 = 2×5.97×10^24 × 6.67×10^-11 / 6.37×10^6 - 9000^2 = 4.40232 ×10^7
b= -2MG = -7.96398 ×10^14
c= (L/m)^2=(v0 R sin α)^2 = 8.2168×10^20

Solve for r:
r = (-b +/- sqrt(b^2-4ac))/2a  = {(7.96398 +/- sqrt( 7.96398^2 - 14.469)) /(2× 4.40232 ) } ×10^7
r = {9.045 +/- 7.947} ×10^6 m

Thus we find the
 - perigee (nearest point of orbit): 1.10 ×10^3 km with speed 26.1 km/s (fictitious), and the
 - apogee (the farthest point): 1.70 ×10^4 km with speed 1.69 km/s

For α =0, we get c=0 and r=-b/a or 0.

[nybble]

I have to say you look great in that color shirt! I found one that looks fairly similar at a consignment shop in Davis Square :)

Glad to see you're doing well!

[neemeshyadav]

Love and Respect from INDIA Professor.

[golamishtiak]

Ah, I did it horribly wrong...🤕 I didn't thought about the ellipse at all :'(
But the problem is actually more nice than i thought.😀 thank you professor, and hope you keep posting problems like this on and on.

[dreadwolf]

Where can I find these problems, Professor?

[rabidromi]

Lots of love and respect to you sir from India

[akashattri]

Hello sir,
I'm Aakash, I'm form India
and i really liked the solution of problem
and i'm an admirer of you, I just completed reading your book (couple of days actually) and i would really like it to be signed by you
I don't want to waste your time, bit still I'm hopefull,
Thanking you.

[ganymedkallisto]

If you start the object with 90° as shown at 13:00, the object doesn't come back at the other side of the planet at distance R and with 9000 m/s speed. The center of the planet still will be at one focus of the ellipse. So, the starting- point will be the periapsis. The apoapsis will be at the other side ("down" in your picture). So, the solution of V2 you get from your equation, is the "trivial" solution for the starting- point.

[dheerajck]

sir can i meet u
am asking this not to get solution to get answer to big problems


it is to have a good base in these ideas so i can think of my own

[apal]

Dear Sir, if we consider this planet has atmosphere then how could we be able to solve this problem?
I am asking you to just mention that how difficult this to solve considering it has atmosphere?

[carpediemyes]

Assume 2 similar masses of radius R, with approx the same size are near each other; then if we fire one away from the other, there will be a critical speed [escape v] at which it will continue to move away. Below that speed it will come back and it is reasonable that it will move in the path of an Ellipse with one of the Foci being the 1st mass. if there is a large difference in size or mass OR if they are in very close proximity, will the 2nd mass still move in an Elliptical path ??? Is the CofG assumption still valid ?? We know it will move in a Parabolic path for Projectile Motion and that satellite motion is really Projectile Motion. So the Elliptical path assumption of this problem seems to me to be a stretch at best and possibly not valid at all.

[architsharma]

Can this problem be solved by relativity??

[europaeuropa]

I wonder how many high school students can honestly say they figured this problem out by themselves
without any outside help and not asking someone else any questions or checking their answers against another person.
Professor, what happens when alpha = 0 ?

[reethanyas]

Hey sir Plss reply when do you come to india

[violetharper]

You’re the only half jewish physicist I’ve heard of, you’re my role model!

[நக்மாசெல்]

Mr.lewin. your physics is good. I wish I should invite you for our Indian breakfast

[yogeshkumar-zvix]

Hello sir I am Yogesh from India, and currently pursuing bsc physics honours from Delhi university, being a physics student looks awkward to ask this question but on this question I am pondering for a couple of weeks sir my question is I have a keen interest to pursue my career in cancer research, for which my friends were telling that I should have a good knowledge of organic chemistry but as I am currently studying Physics so sir can you guide me what should be my approach for heading towards my goal because cancer research is my biggest dream but being a physics student always pulls me back and again forced me to think over whether I will be able to succeed in it or not . Because I should have a chemistry background for this . It will be very helpful if you just guide me a little bit .

[soumyadipdas]

Sir can u please explain why the mass will not colloid the earth?I don't understand the explanation ..why the mass will not colloid the earth!