PHSC108_Hmwk6.pdf - PHSC 10800: Earth as a Planet Homework...

PHSC 10800: Earth as a Planet
Homework #6
Show all work for full credit.
Assigned: November 12, 2018
Due for Full Credit: November 19, 2018
1) Planet Velocities and Energy (33 pts)
We talked about how planet formation involves the collisions of bodies
(planetesimals, embryos) leading to the growth (and heating) of a planet. Let’s think
about the velocities and energies involved here.
a) The speed of a body in its orbit around the Sun is given by the equation:
V
2
= GM
⦿
[(2/r) - (1/a)]
Here V is the speed of the body in m/s, G is the gravitational constant, M
⦿
is the mass of
the Sun in kilograms, a is the semi-major axis of the body in meters, and r is its current
distance from the Sun in meters (remember, in an elliptical orbit, the distance between
the Sun and the body is always changing). Assume you have an asteroid with a semi-
major axis of 3 AU and a perihelion of 1 AU (so it’s potentially an Earth crossing orbit).
What would its speed be at 1 AU? If the Earth’s orbital speed is 30 km/s, what is the
lowest velocity at which this asteroid could possibly impact the Earth (what is the
difference in orbital velocity, if both bodies are orbiting in the same direction)? What if it
was a comet, which has a semi-major axis of 20 AU and a perihelion of 1 AU? ( Here
we’re ignoring the acceleration from Earth’s gravity to keep things simple.)
b) The accretion of planets can involve very large, energetic impacts. The impact
that formed the Moon is typically assumed to have involved the collision of a Mars-sized
impactor (about 10% the mass of the Earth) with the Earth at a velocity of 10 km/s. If
the kinetic energy of a moving body is given by E
kinetic
=(1/2)mv
2
where m is its mass and
v its speed, what was the energy of the kinetic energy of the impactor? Give your
answer in Joules (J) where 1 J = 1 kg m
2
/s
2
.