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You should really try to solve this yourself first, in order to maximize the value in doing that. If you read this solution, please understand each step before proceeding to the next step, otherwise the lesson will be lost to you.
To determine the required elevation of an observer in order that he may be able to see an object on the earth thirty miles away, assuming the earth's radius is 3956 mi and the earth is a smooth sphere, first draw the triangle involved.
This is a right triangle, where the hypotenuse is the radius of the earth plus the elevation of the observer. One side is the radius of the earth to the object. The other side is the line of sight distance from the observer to the object, which is greater than 30 miles. That line of sight is tangent to the earth's circumference, at the point of the object, so the angle of line of sight relative to the radius at the object is 90 degrees.
The angle at the center of the earth is 360 degrees times 30 miles divided by the circumference of the earth, 2 pi 3956, or 24856, which is an angle of 0.4345 degrees.
The hypotenuse of a right triangle with one angle of 0.4345 degrees and side of 3956 is 3956 divided by cosine 0.4345 degrees, or 3956.1138 miles.
Subtract the radius, 3956 miles, and you get 0.1138 miles, or 601 feet.
Not asked, but answered for completeness, is that the line of sight distance is hypotenuse times sine 0.4345 degrees, or 30.0005751 miles, or 3.04 feet more than 30 miles.
What else can I help you with?
How do you find the angle of depression?
The angle below horizontal that an observer must look to see an object that is lower than the observer. Note: The angle of depression is congruent to the angle of elevation (this assumes the object is close enough to the observer so that the horizontals for the observer and the object are effectively parallel; this would not be the case for an astronaut in orbit around the earth observing an object on the ground).
What is called the angle between an object and the horizon?
The angle between an object and the horizon is referred to as the "angle of elevation" when the object is above the horizontal line, and the "angle of depression" when the object is below it. These angles are commonly used in trigonometry and surveying to determine heights and distances. The angle of elevation is measured from the observer's eye level upward to the object, while the angle of depression is measured downward from the observer's eye level to the object.
What is the mathematical definition of angle of elevation?
The angle of elevation is defined as the angle formed between a horizontal line and the line of sight to an object that is above the horizontal line. It is measured from the observer's eye level up to the object. In mathematical terms, it can be expressed using trigonometric functions, where the tangent of the angle is the ratio of the opposite side (the height of the object) to the adjacent side (the distance from the observer to the base of the object).
What is the angle of declination and how do you use it?
An angle of declination is relevant when an observer is at a higher altitude than the object being observed. It is the angle made by the line of sight with the horizontal. Suppose this is angle x. Then if the altitude of the observer is known to be h, then line-of-sight distance to the object is h*sin(x). The object is h*tan(x) from the point below the observer at the level of the object.Conversely, if the line-of-sight distance from the object to the observer or the horizontal distance to the point directly below the observer is known, it is possible to calculate the height of the observer.
What is the equation for height in physics?
Height of an object = (elevation of its top) - (elevation of its bottom)
How do you find the angle of depression?
The angle below horizontal that an observer must look to see an object that is lower than the observer. Note: The angle of depression is congruent to the angle of elevation (this assumes the object is close enough to the observer so that the horizontals for the observer and the object are effectively parallel; this would not be the case for an astronaut in orbit around the earth observing an object on the ground).
What is called the angle between an object and the horizon?
The angle between an object and the horizon is referred to as the "angle of elevation" when the object is above the horizontal line, and the "angle of depression" when the object is below it. These angles are commonly used in trigonometry and surveying to determine heights and distances. The angle of elevation is measured from the observer's eye level upward to the object, while the angle of depression is measured downward from the observer's eye level to the object.
What is the mathematical definition of angle of elevation?
The angle of elevation is defined as the angle formed between a horizontal line and the line of sight to an object that is above the horizontal line. It is measured from the observer's eye level up to the object. In mathematical terms, it can be expressed using trigonometric functions, where the tangent of the angle is the ratio of the opposite side (the height of the object) to the adjacent side (the distance from the observer to the base of the object).
What is an azimuth and elevation?
These are terms used in surveying and astronomy (usually quotes in angles) to denote the apparent position of an object in the sky based on a specific obervation point or an observer on the Earth's surface. See the following link for an example: http://searchcio-midmarket.techtarget.com/sDefinition/0,,sid183_gci838808,00.html
How the apparent motion of a object depend on the observer motion?
The apparent motion of an object can vary depending on the motion of the observer. This is due to the concept of relative motion, where the perception of an object's movement is influenced by the observer's own motion. For example, if the observer is moving towards an object, the object may appear to move faster than if the observer is stationary.
If you propel an object from ground level with a known trajectory and force and then propel the same object from a higher altitude with the same trajectory and force which object will travel further?
Assuming that in each case the object will not be acted on my any other force and that the object will not have to land at an elevation different from the elevation it left from, and assuming that the object will not encounter any obstacles-- the object will travel further if when launched from the higher altitude because it will encounter less gravity and less resistence from the air it travels through.
How does apparent motion of an object depend on the observer and in motion?
The apparent motion of an object depends on both the observer's perspective and the motion of the object itself. As the observer moves, their angle of view and distance from the object change, altering how the object appears to move relative to them. In addition, the speed and direction of the object's actual motion will impact how it appears to move to the observer.
From the perspective of a stationary observer, does time for an object in motion relative to the observer"?
Yes, time for an object in motion relative to a stationary observer appears to pass slower.
How is the apparent size of an object affected by its distance from the observer?
The apparent size of an object decreases as it moves farther away from the observer. This is because the angle that the object subtends at the observer's eye decreases as the distance increases, making the object appear smaller.
How is a UFO created?
When a strange object flies into the scope of an observer, and the observer is incapable of finding a suitable moniker for the object it is given the paradoxical identity of an Unidentifieed Flying Object.
How is the size of an object affected by its distance from an observer?
As an object moves farther away from an observer, it appears smaller due to perspective, which causes a decrease in angular size. This change in size is a result of the viewing angle between the observer and the object decreasing with distance.
What increases as an object comes closer to an observer?
As an object comes closer to an observer, the object's apparent size increases, allowing the observer to see more details. The object's brightness may also increase due to a larger portion of light being collected by the observer's eye or camera. Additionally, the parallax effect becomes more pronounced, providing a sense of depth and movement to the object.
