(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A asteroid is approaching our solar system from a great distance (r->infinity) and is headed straight for the Sun. Because of the gravitational pull of the Sun the asteroid is constantly accelerating.

Radius of Earth's Orbit around the Sun ([tex]R_{e}[/tex]) 1.5*10^11

Venus is given as 3/4 of this.

Sun's mass (M) 2.0 * 10^30kg

a)What is the speed (v) of the asteroid, which was originally stationary at a very great distance, as it comes as close to the Sun as the Earth orbits?

b)How long does the asteroid take to travel from crossing the orbit of the Earth around the Sun to crossing Venuse's orbit.

2. Relevant equations

[tex]F= \frac{G*mM}{r^2}[/tex]

[tex]E_{p}=m*g*r[/tex]

[tex]E_{k}=\frac{m*v^2}{2}[/tex]

[tex]Work=\int F ds[/tex]

3. The attempt at a solution

a)Energy is conserved. The sum of potential and kinetic energy is constant. noting that g of course changes with r (distance from the Sun)

g=grav. const. m*M/r^2

[tex]\frac{m*v^2}{2} = \int^{\infty}_{0}\frac{G*m*M}{r^2}dr - \int^{R_{e}}_{0}\frac{G*m*M}{r^2}dr[/tex]

[tex]v^2 = 2*(\int^{\infty}_{0}\frac{G*M}{r^2}dr - \int^{R_{e}}_{0}\frac{G*M}{r^2}dr)[/tex]

[tex]v^2 = 2*G*M*(\int^{\infty}_{0}\frac{1}{r^2}dr - \int^{R_{e}}_{0}\frac{1}{r^2}dr)[/tex]

[tex]v^2 = 2*G*M*r_{E}^-1[/tex]

v=4,22 m/s?

I'm posting this because the units don't add up so I must have made a mistake either in the set up or in the integration itself (which I don't think I did). I translated the text from my own language to English so if the instructions or notation doesn't make sense please ask. Thanks for your help and patience! :)

b)Hm this one proved to be more difficult. Its no trouble at all to see that I more or less basically have a function of v(r).

[tex]v=\sqrt{\frac{2*G*M}{r}}[/tex]

r obviously changes from Earth's Orbital radius to the Orbital Radious of Venus (given as 3/4 of Earths).

So I'm going to mark travelled distance as s, which is of course:

[tex]s=\int^{t}_{0}v(t)dt[/tex]

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# Homework Help: A asteroid approaches the Sun

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