Universal gravity - check my answer

1. Mar 11, 2009

tnutty

1. The problem statement, all variables and given/known data

Mercury's orbital speed varies from 38.8km at aphelion to 59.0km at perihelion.

If the planet is 6.60*10^10m from the Sun's center at aphelion, how far is it at perihelion?

2. Relevant equations

v = sqrt(GM/r) (speed,circular orbit)

3. The attempt at a solution

V_a = velocity of aphelion = 38.8km = 3.88 * 10^4m
V_p = velocity of perihelion = 59.0km = 5.9 * 10^4m

V_a = sqrt(GM/r)

3.88 * 10^4m = sqrt(GM/r_a)
and solving for M gives me (I think)

M = (3.88 * 10^4m*r_a)^2 / G

V_b = sqrt(GM/r_p) ; where W_b = 5.9 * 10^4m

solving for r_p gives :

r_p = (3.88 * 10^4m *r_a ) / 5.9 * 10^4m

; where r_a = 6.60*10^10m;

and

r_p = 4.60 * 10^10

is this right?

2. Mar 11, 2009

AEM

Since Mercury travels in an ellipse you shouldn't be using an equation for velocity in a circular orbit. What you want to do is to use the fact that total energy is conserved. That is,

KE (Perihelion) + PE (perihelion) = KE (aphelion) + PE (aphelion)

Once you express these energies in their proper form, you will see that you have only one unknown -- the distance at perihelion.

3. Mar 11, 2009

tnutty

Is this the right equation ;

KE (Perihelion) + PE (perihelion) = KE (aphelion) + PE (aphelion)

0 - GMm/ r_e = 1/2 mv^2 - GMm/r_a

and solving for r_e I get (check it )

r_e = GM / (1/2 v^2) - r_a

and plug in the numbers now, if this is correct

Last edited: Mar 11, 2009
4. Mar 11, 2009

tnutty

anyone?

5. Mar 12, 2009

LowlyPion

How did you get that KE at perihelion was 0?

6. Mar 12, 2009

tnutty

ok wait here is what i think is correct,

KE (Perihelion) + PE (perihelion) = KE (aphelion) + PE (aphelion)

1/2*mv^2 - GMm/ r_e = 1/2 mv^2 - GMm/r_a

= (check this below?)

r_e = -GMm / (1/2mV_a^2 - GMm/r_a -1/2mV_p^2 )

r_e = -GM / (.5V_a^2 - GM/r_a - .5V_p^2 )

r_e = 4.598 * 10^10

7. Mar 12, 2009

tnutty

I guess that is correct but I have 1 question :

KE (Perihelion) + PE (perihelion) = KE (aphelion) + PE (aphelion)

it does not matter if

the above formual is switch right, the perihelion on the right side and
the aphelion on the left?