Question
SHOW WORK!
The first proposal for how an artificial satellite would work is attributed to Isaac Newton. He reasoned that due to Earth's curvature, one could create an artificial satellite by launching an object horizontally at just the right speed. The object would follow projectile motion and fall toward the Earth's center. Due to the surface of the Earth being curved, the surface would 'move away' from a horizontal trajectory. If the object's rate of falling toward the Earth's surface matched the rate at which the surface is curving away from a straight trajectory, the object would be in a constant state of falling but never coming any closer to the surface of the Earth. A rough illustration of the effect is shown below. For example, near the surface of the Earth, one can find that the object has to fall vertically about 5 meters for every 8000 meters of horizontal motion in order to maintain a circular orbit around the Earth. This model works well even for satellites that are far from, but maintain a circular orbit around, the planet they are orbiting. Assume a satellite is circling around Planet X at a radius of 11.0 times 103 km from the center of the planet. For every 7.5 kilometers, how many meters does it have to fall towards Planet X in order to maintain a circular trajectory? meters for every 7.5 kilometers to maintain a circular trajectory. [Comment: this problem can be solved in at least two completely different ways. One way uses radial acceleration and Newton's universal law of gravity, but the number of "missing' variables from the problem text can be intimidating - a lot of unknowns will cancel with the proper substitution(s). The other way is to reason from pure geometry of the trajectory, but the setup + trigonometry can seem a bit tricky.]
Explanation / Answer
r^2 = x^2 + y^2
dy = r - y
r-y =1000*
11.0E3-sqrt(11.0E3^2 - 7.5^2)=2.56 m
