Calculating Central Force IV Velocities with m, L, a and e

[Nusc]

A planet of mass m follows an elliptic path around the Sun with a semi-major axis a and an eccentricity e, using the polar form for the radial position of the planet with respect to the sun:

a) Find both the radial component of the velocity Vr and the angular component of the velocity V theta in polar form

r(t) = r <er>
V(t) = r* <er> + r theta* <e theta>

b) Show that when the planet is located at the perihelion (r=rmax) and at the aphelion (r=rmax) the velocity of the planet has no radial component. Compute the angular component of the velocity at these two points.

r = rmax = a(1+e)
r = rmin = a(1-e)

r* = 0

Vmin = 0 <er> + a(1+e) theta* <e theta>
Vmax = 0 <er> + a(1-e) theta* <e theta>

c)Show that when the orbit is circular, the radial component of the velocity vanishes for all time and the angular component of the velocity is always constant. Give your answers to the above in terms of the constants, m, L, a and e.

How do I start this one provided that the 2 parts above are correct?

 

[anathema]

To begin, we can use the equation for the radial position of the planet in polar form: r(t) = r <er>, where r is the distance from the planet to the Sun and <er> is the unit vector in the radial direction.

a) To find the radial component of the velocity, we can take the derivative of r(t) with respect to time: Vr(t) = r* <er>. This gives us the velocity in terms of the unit vector <er>, which represents the direction of the velocity.

Next, we can find the angular component of the velocity by taking the derivative of r(t) with respect to the angle theta: V theta(t) = r theta* <e theta>. This gives us the velocity in terms of the unit vector <e theta>, which represents the direction of the velocity perpendicular to the radial direction.

b) At the perihelion, the distance from the planet to the Sun is rmax = a(1+e). Plugging this into the equation for the radial component of the velocity, we get Vr = rmax* <er> = 0, since rmax* is equal to 0. This means that at the perihelion, the velocity has no radial component.

To find the angular component of the velocity at the perihelion, we can plug in rmax = a(1+e) into the equation for the angular component of the velocity: V theta = a(1+e) theta* <e theta>. This gives us a non-zero velocity in the direction perpendicular to the radial direction.

Similarly, at the aphelion, the distance from the planet to the Sun is rmin = a(1-e). Plugging this into the equations for Vr and V theta, we get Vr = 0 and V theta = a(1-e) theta* <e theta>.

c) When the orbit is circular, the eccentricity e = 0, and the distance from the planet to the Sun is constant at a. This means that r* = 0, and the radial component of the velocity is always 0.

To find the angular component of the velocity, we can plug in e = 0 into the equation for V theta: V theta = a theta* <e theta>. This gives us a constant velocity in the direction perpendicular to the radial direction.

In terms of the

 

[quicknote]

To start, we can use the definition of angular momentum (L) to relate it to the orbital parameters. The angular momentum of a planet in orbit around the sun is given by L = m*r^2*theta*, where m is the mass of the planet, r is the distance from the planet to the sun, and theta is the angular velocity. We can also express the angular velocity in terms of the semi-major axis (a) and eccentricity (e) as theta = sqrt(Gm/a^3)*(1-e^2)^(-3/2), where G is the gravitational constant.

a) To find the radial component of the velocity, we can take the derivative of the radial position with respect to time (t) and multiply by the unit vector <er>. This gives us Vr = r* <er>. To find the angular component of the velocity, we can take the derivative of the angular position with respect to time and multiply by the unit vector <e theta>. This gives us V theta = r theta* <e theta>.

b) At the perihelion, r = rmax = a(1+e), so the radial component of velocity is 0, as r* = 0. The angular component of velocity can be calculated using the formula for theta given earlier. So at the perihelion, V = a(1+e) theta* <e theta>. Similarly, at the aphelion, r = rmin = a(1-e), so the radial component of velocity is 0, and the angular component of velocity is V = a(1-e) theta* <e theta>.

c) When the orbit is circular, e = 0, so the angular velocity becomes theta = sqrt(Gm/a^3), which is a constant. This means that the angular component of velocity, V theta, is also constant. Additionally, since the orbit is circular, r = a, so the radial component of velocity becomes Vr = a* <er>, which is also constant. This means that the radial component of velocity vanishes for all time.

In terms of the constants, we can rewrite the angular component of velocity at the perihelion and aphelion as V = (L/m)*(1+e) <e theta> and V = (L/m)*(1-e) <e theta>, respectively. And for a circular orbit, the angular component of velocity