How Far Does the Moon Fall Toward Earth in One Second?

[eunhye732]

The center-to-center distance between Earth and Moon is 384400 km. The Moon completes an orbit in 27.3 days.
(b) If gravity were switched off, the Moon would move along a straight line tangent to its orbit, as described by Newton's first law. In its actual orbit in 1.00 s, how far does the Moon fall below the tangent line and toward the Earth?

part (a) was asking for the Moon's orbital speed which was 1024 m/s. for part (b) my question is I don't know what equation to use to solve for this. All i know is that I need to use centripetal acceleration but when?
thanks and i hope you can help~!

 

[mezarashi]

The second part is a geometry problem.
Consider a circle with a tagent line drawn at one point. Now after 1.0 s, the moon (under gravity) moves through an angle of theta. I truly think this needs a diagram, so I've attached one now. We want to solve for y.

http://img400.imageshack.us/img400/503/745636345ew.gif

 

[quicknote]

To solve for the distance the Moon would fall below the tangent line towards the Earth, we can use the equation for centripetal acceleration, which is given by a = v^2/r, where v is the orbital speed and r is the distance between the Moon and the Earth.

In this case, we know the orbital speed (1024 m/s) and the distance between the Earth and Moon (384400 km or 384400000 m). We can convert the distance to meters for consistency.

Plugging in the values, we get:
a = (1024 m/s)^2 / (384400000 m)
= 0.0000027 m/s^2

This is the acceleration towards the Earth due to gravity. To find the distance the Moon would fall in 1.00 s, we can use the equation for displacement, given by d = 1/2 * a * t^2, where a is the acceleration and t is the time.

Plugging in the values, we get:
d = 1/2 * (0.0000027 m/s^2) * (1.00 s)^2
= 0.00000135 m

Therefore, the Moon would fall approximately 0.00000135 meters towards the Earth in 1.00 second if gravity were switched off. This may seem like a small distance, but over time it would accumulate and cause the Moon to eventually collide with the Earth. This highlights the importance of the universal law of gravitation in keeping celestial bodies in their orbits.