So this is the problem:
A comet is in an elliptical orbit around our Sun. At its point of closest approach, it is moving at 6.3x10^4 m/s and is 4.2x10^10m away from the sun. When it is at the farthest distance, 7.5x10^12m away, how fast is it going?
This is what I did
K1+U1=K2+U2 (since energy should be conserved)
so...
1/2m1v1^2 + -Gm1m2/r1 = 1/2m1v2^2 + -Gm1m2/r2
the m1 cancel since all of them have it
1/2v1^2 + -Gm2/r1 = 1/2v2^2 + -Gm2/r2
then I rearranged to get v2 by itself
1/2v1^2 + -Gm2/r1 + Gm2/r2 = 1/2v2^2
*2 *2
(v1)^2 +-2Gm2/r1 + 2Gm2/r2 = (v2)^2
(v1)^2 + 2Gm2/r2 -2Gm2/r1 = (v2)^2
simplified it
(v1)^2 + (2Gm2 (1/r2 - 1/r1))= (v2)^2
and then square root for the final equation to be
((v1)^2 + (2Gm2 (1/r2 - 1/r1)))^.5 = (v2)
Then I plugged in the numbers
v1 = 6.3x10^4
G= 6.67x10^-11
r2= 7.5x10^11
r1= 4.2x10^10
m2 (mass sun) = 2x10^30
But when I plugged all this in to solve for v2 I got a negative number under the square root. Where did I go wrong? Is my math right? Did I set up the equations right?
How about we write + - as just -
:)
whoops typo = ten times further
r2= 7.5x10^11
should be
r2= 7.5x10^12
Your math and equations look correct, but there seems to be a typo in the value of r2 that you provided. In your question, you stated that the farthest distance is 7.5x10^12m away, but you wrote 7.5x10^11m in your calculation.
Let's double-check the calculations using the correct value of r2 = 7.5x10^12m.
Plugging in the values:
v1 = 6.3x10^4m/s
G = 6.67x10^-11 N(m/kg)^2
r2 = 7.5x10^12m
r1 = 4.2x10^10m
m2 (mass of the sun) = 2x10^30 kg
Now we can substitute these values into the equation:
((v1)^2 + (2Gm2 (1/r2 - 1/r1)))^.5 = (v2)
((6.3x10^4)^2 + (2 * 6.67x10^-11 * 2x10^30 * ((1 / 7.5x10^12) - (1 / 4.2x10^10))))^.5 = (v2)
Plugging the values in the equation:
((3.969x10^9) + (2 * 6.67x10^-11 * 2x10^30 * ((1 / 7.5x10^12) - (1 / 4.2x10^10))))^.5 = (v2)
Evaluating the expression within the square root:
((3.969x10^9) + (2 * 6.67x10^-11 * 2x10^30 * (1.3333333x10^-14 - 2.3809524x10^-14)))^.5 = (v2)
Simplifying further:
((3.969x10^9) + (2 * 6.67x10^-11 * 2x10^30 * (-1.0476191x10^-14)))^.5 = (v2)
((3.969x10^9) - (2.8013360004x10^6))^.5 = (v2)
(3.965198999x10^9)^.5 = (v2)
v2 ≈ 6.292x10^4 m/s
Therefore, the correct answer is approximately 6.292x10^4 m/s when the comet is at the farthest distance.