- #1

Mark Brewer

- 38

- 4

## Homework Statement

A start to solving this question would be much appreciated.

Question: An asteroid from the Kuiper belt (approximately 30 AU from the Sun) is knocked out of its circular orbit and begins to fall straight inwards towards the Sun. We want to know how long it will take to get here and how fast it will be moving when it does.

Using the conservation of energy we have:

E =1/2(mv^2) - (GMm/r)

where E is the total energy of the system, m is the mass of the asteroid v is its velocity, G is Newton's constant, M is the mass of the Sun, and r is the distance from the asteroid to the Sun. As the asteroid begins its journey very far away (compared to where it ends up) we may approximate its initial potential energy by zero. Also, assuming (rather unrealistically) that the asteroid starts falling from rest, we may take its initial kinetic energy to be zero as well, so that E = 0

a) Write down the differential equation for r(t) giving the position of the asteroid as a function of time. Pay close attention to signs.

b) Solve the differential equation for t(r), the time as a function of position.

c) Plug in the numbers to find out how long it will take the asteroid to reach Earth orbit. Take G = 6.67 x 10^-11 (Nm^2) / (kg^2) and M = 2 x 10^30 kg.

d) How fast will it be moving when it crosses Earth's orbit (in km/s)?

e) Assuming the asteroid has a mass m = 9 x 10^20 kg (approximately equal to that of Ceres), compute the kinetic energy released if it were to collide with the Earth. Remark: For comparison the energy released by the asteroid that wiped out the dinosaurs was estimated to be around 4 x 10^23 J.

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## The Attempt at a Solution

My attempt to question #1:E = 0

r = (v^2) / 2GM (conservation of energy equation)

r = r1 ((v+v1) / 2) t (constant acceleration)

r = ((v^2) / GM(v+v1) (1./t))

dr/dt = ((v^2) / GM(v+v1) (1/t^2))

Am I on the right track?

My attempt to question #2:

dr/dt = ((v^2) / GM(v+v1) (1/t^2))

by separable variables

(t^-2)dt = (GM (v2 + v) / v^2)dr

by integration I get

* = (GM (v2 + v) / v^2) r

t = cube root ((v^2/(GM(v2+v)r) +c)

Am I still on the right track?

My attempt to question #3:

t = 0

I know I'm lost now.