Calculate Mercury's Orbital Speed and Distance: Universal Gravity Equation

[tnutty]

Homework Statement

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?

Homework Equations

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

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?

 

[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.

 

[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

 

[tnutty]

anyone?

 

[LowlyPion]

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

How did you get that KE at perihelion was 0?

 

[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

 

[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?