Question

An asteroid of mass m is in a circular orbit of radius r around the Sun with a speed v. It has an impact with another asteroid of mass M and is kicked into a new circular orbit with a speed of 1.5 what is the radius of the new orbit in terms of r?

Short answer

The radius of the new orbit in terms of \({\rm{r}}\) is \({r_2} = \frac{4}{9}r\)

Step-by-step solution

Find the radius of the new orbit in terms of r

If an object is moving in a circle of the radius \({\bf{r}}\) with constant speed \({\bf{v}}\), then the motion is a uniform circular motion. It has a radial acceleration \({a_R}\) that is directed radially toward the center of the circle, which is called radial acceleration, given by:

\({a_R} = \frac{{{v^2}}}{r}\)

Also, by Newton's law of universal gravitation, every particle in the universe attracts every other particle with force proportional to the product of their masses and inversely proportional the square of the distance between them. Therefore,

\(\begin{aligned}{F_G} &= G\frac{{m{m_{Sun}}}}{{{r^2}}}\\ &= m{a_{Sun}}\end{aligned}\)

Determination of the radial acceleration

Now, the radial acceleration is given by:

\(\begin{aligned}{a_R} &= \frac{{{v^2}}}{r}\\ &= G\frac{{{m_{Sun}}}}{{{r^2}}}\end{aligned}\)

Rearrange and solve for the speed to get:

\(\begin{aligned}{v_2} &= \sqrt {\frac{{G{m_{Sun}}}}{{{r_2}}}} \\ &= 1.5v\\ &= 1.5\sqrt {\frac{{G{m_{Sun}}}}{r}} \end{aligned}\)

Rearrange and solve for the new radius to get:

\(\begin{aligned}{r_2} &= \frac{r}{{{{\left( {1.5} \right)}^2}}}\\ &= \frac{4}{9}r\end{aligned}\)

So, the radius of the new orbit is \(\frac{4}{9}r\).