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Redbelly98 said:PF guidelines are that HW posters should show an attempt at solving the problem before giving help.
One implication of this is, HW answerers should not give help until the asker has made some attempt. Please let the asker show what they know first, before helping out.
I'm trying to understand this. I understand the last bit, but I'm just trying to really understand it in deep... nothing about your explanation, I'm just blank and tired. So if anyone can make it more clear, I would really appreciate it. And thanks very much too!Almanzo said:In space, a circle touching the bullet's track on the inside would have a much larger radius (hence less curvature) than a circle touching the ball's track on the inside. However, the problem is to be solved in spacetime. Now, the bullet takes less time to complete its track than the ball (which is evidenced by its rising and falling over less distance in the same gravity field to arrive on the same spot (in space) as the ball. The spacetime track of the ball is therefore much longer than the spacetime track of the bullet. If one uses the Euclidean metric, that is, but I think one must in this case.
The radius of a circle segment equals the square of the length of half its chord divided by its height. The length of the chord is nearly equal to the time difference times c, and this time difference can be calculated (approximated) by classical Newtonian means. The squares of the time differences should be in the same proportion as the heights, as both objects are rising and falling in the same gravity field. Therefore, the radii should be equal, and so must the curvatures be equal.
KiyoEtAlice said:...I've tried using simple geometry and I think it's worng, but I'm blank about what formula I should use.
Redbelly98 said:Normal geometry is the way to go here. Draw a figure with the circular arc, and the circle's center (way below the arc)
There are two options for working this out:
Equation of a circle (in terms of radius and coordinates of the center)
or
Trigonometry
KiyoEtAlice said:Okay, now I'm stuck agian. I'm not sure which formula to use to calculate it, or how to calculate it :(
Almanzo said:I am unsure about the source of the confusion.
There are three things to be considered:
1. To calculate the time needed to fall from a certain height in Earth gravity.
2. To calculate the length of a line segment in spacetime.
3. To calculate the radius of a circle, if a chord and and height of a circle segment are known.
The first is classical mechanics: vertical speed increases from zero to a certain speed. The average vertical speed is half the vertical speed on arrival. The time of fall is the height divided by the average vertical speed. It is also final vertical speed divided by g, where g is local gravity (9.81 m/s2). Therefore height fallen is proportional to the square root of time elapsed.
The second is special relativity. The timelike component is c times the time elapsed. The spacelike component is the length of the trail in space. The spacelike component, here is tiny compared to the timelike component; therefore the size of the segment is nearly the size of its timelike component.
The third is elementary geometry. A rectangular triangle can be drawn, having half the chord as short side, radius minus height as long side, and radius as hypothenusa. A second such rectangle has height as the short side and half the chord as the long side. The two are similar; hence height/half chord = half chord/radius.
Note that the distance to the horizon (on an airless world) is the geometric mean of the height on which one is standing and the radius of the world.
Curvature is a measure of how much a curve deviates from a straight line. It is defined as the rate of change of the tangent of a curve at a given point.
Curvature is typically calculated using a mathematical formula that involves the radius of the curve and the angle at which the curve changes direction. It can also be calculated using calculus by finding the second derivative of the curve's equation.
Curvature is typically measured in units of curvature, such as radians per meter or degrees per kilometer. It can also be measured in terms of the radius of curvature, which is the distance from the curve to its center.
Curvature can be seen in many natural and man-made structures, such as bridges, roads, and roller coasters. It is also present in nature, such as in the shape of waves, tree branches, and animal horns.
Curvature plays a role in many aspects of our daily lives, from the design of buildings and infrastructure to the movement of objects and vehicles. Understanding curvature can also help us navigate and understand the world around us, such as reading maps and interpreting geometric shapes.