Gravitation

Question 1

what is the virial theroem

Answer 1

2k + U = 0
the is only for gravitationally bound bodies. k and U are time averages

Question 2

describe the properties of orbits for each conic section type

Answer 2

elliptical/circular: total energy is negative, gravitationally bound

parabolic: total energy = 0, unbound

hyperbolic: total energy is positive, unbound

Question 3

elliptical orbit velocity ratio equation

Answer 3

vₚ/vₐ = rₐ/rₚ = (1+e) / (1-e)
e: eccentricity
Vₚ : perihelion velocity
rₚ: distance to sun at perihelion
ₐ: aphelion equivalent

Question 4

what does firing the rocket to propel in the direction of travel do for a rocket

Answer 4

it moves the rocket to an orbit for a larger semi-major axis but this reduces the average orbital speed

Question 5

angular momentum formula

Answer 5

L = m (r x v)
m: mass
r: distance to sun
v: velocity
x: cross product

Question 6

what’s a Hohmann transfer

Answer 6

the most energy-efficient way of transferring orbits. It puts the rocket in an elliptical orbit. (uses the least fuel). example: Earth to Venus. launch with Earth at the aphelion and land with Venus at the perihelion of the rocket’s orbit. (other way around if going to a planet further from the sun)

Question 7

how do gravitational slingshots work

Answer 7

they are used to save energy when transferring orbits. When flying past a planet, a spacecraft can gain or lose energy (via angular momentum) by transferring it to/from the planet.

Question 8

what’s the effective potential

Answer 8

the sum of the gravitational potential and the rotational energy.

rotational energy comes from:
F꜀ = mω²r r̂
U꜀ = ∫F꜀ dr = - ½mω²r² r̂ (the -r̂ means its in the outward direction)

hence:
U = -GMm/r - ½mω²r²
U: effective potential

for per unit mass potential remove the small m.

Question 9

Explain how a rocket’s speed changes using a gravitational slingshot

Answer 9

Initial speed, v, toward the planet (planet’s rest frame). The angle of deflection is α.

In the sun’s frame, the planet has a speed, u, and angle, θ, to the vertical.

hence in the sun’s rest frame, the rocket has speed v + ucos(θ) in the vertical and usin(θ) in the horizontal

after the slingshot in the sun’s frame:
vertical component: vcos(α) + ucos(θ)
horizontal component: vsin(α) + usin(θ)

example: (if θ=90 and α=90)
initial speed = sqrt(v+u)
final speed = u+v