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

In summary, 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?
  • #1
tnutty
326
1

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?
 
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  • #2
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
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:
  • #4
anyone?
 
  • #5
tnutty said:
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?
 
  • #6
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
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?
 

1. What is universal gravity?

Universal gravity, also known as Newton's Law of Gravitation, is a fundamental physical law that describes the attractive force between two objects with mass. It states that the force of gravity is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

2. How did Isaac Newton discover universal gravity?

Isaac Newton discovered universal gravity in 1687 through his famous thought experiment of the falling apple. He then developed his theory of gravitation by combining it with the laws of motion and publishing it in his book "Principia Mathematica".

3. What is the difference between universal gravity and general relativity?

Universal gravity is a classical theory that describes the force of gravity between two objects with mass. General relativity, on the other hand, is a modern theory of gravity that describes it as a curvature of space-time caused by the presence of mass and energy.

4. What are some real-world applications of universal gravity?

Universal gravity has many practical applications in our daily lives, such as predicting the motion of planets and other celestial bodies in our solar system, calculating the trajectory of satellites, and determining the weight of an object on different planets.

5. Is universal gravity still a valid theory?

Yes, universal gravity is still considered a valid theory in physics, as it accurately describes the force of gravity in most situations. However, it does not fully explain certain phenomena, such as the rotation of galaxies, which is where the theory of general relativity comes into play.

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