Question

A moon worm is orbiting an asteroid in a circular path with radius r and speed v. If the moon worm decides to double its speed, how will it have to change the radius of its path so that it continues to orbit the asteroid in a perfect circle?

Answer 1

Answer on Question #50550 – Physics – Mechanics | Kinematics | Dynamics

Question.

A moon worm is orbiting an asteroid in a circular path with radius $r$ and speed $v$. If the moon worm decides to double its speed, how will it have to change the radius of its path so that it continues to orbit the asteroid in a perfect circle?

Solution.

Let remember the Newton’s second law:

$$\sum F = m a$$

In our case, the acceleration is the centripetal.

$$a = a_{c}$$

By definition the centripetal accelerations is equal to:

$$a_{c} = \frac{v^{2}}{R}$$

To continue to orbit the asteroid in a perfect circle it must be satisfied:

$$\sum F = const$$

But in our case, $m = const$.

Therefore,

$$a_{c} = const \rightarrow \frac{v^{2}}{R} = const$$

So, if we double the speed $v = 2v_{0}$, then:

$$\frac{v^{2}}{R} = \frac{v_{0}^{2}}{R_{0}} = const$$ $$\frac{(2v_{0})^{2}}{R} = \frac{v_{0}^{2}}{R_{0}} \rightarrow R = 4R_{0}$$

Thus, to double the speed it needs to increase 4 times the radius.

Answer.

To increase 4 times: $R = 4R_{0}$

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