PHY100 Homework 3

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Astronomy

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Oct 30, 2023

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PHY100 Homework 3 Weight 8% 1. Watch the following video by Terence Tao on Cosmic Distance Ladder and write a 200- word report on your favorite part of the talk. The talk is available at: https://www.youtube.com/watch?v=7ne0GArfeMs . His slides for the talk are available here: https://terrytao.files.wordpress.com/2010/10/cosmic-distance-ladder2.pptx (20 pts) Terence Tao's discussion about how technology isn't necessarily required to solve problems in the cosmos was my favorite part of the talk. Simple geometry, for example, is all that is required to determine measures to answer calculations, and this was done many years ago by ancient Greek philosophers. Aristotle deduced that the earth was round by using indirect observations of how a shadow is always circular and the shadow projected by the earth onto the moon. Eratosthenes was another Greek who calculated the radius of the Earth by looking at the sun and measuring the distances between cities on the planet. It's amazing to learn that the Greeks used indirect observations and elliptical arcs rather than technology to determine the forms and distances of celestial objects. The fact that several of the observations turned out to be almost exact suggests that mathematical relationships have helped us understand space more deeply. Taking everything into account, the fact that individuals were able to calculate cosmic facts using only their minds and not technological advances illustrates just how amazing a train of thought can be. 2. Principle of inertia and gravity. Consider a boat coasting on a quiet lake with a ball hung at the top of its mast. At a certain time ball is let go. Plot the trajectory of the ball and the boat as seen by: a. A person on the boat (5 pts) The ball would seem to fall straight down to someone that on the boat. Due to the fact that, prior to the ball being dropped, both boat and ball are going at the same speed. When the ball is released, it accelerates due to gravity and moves with the boat horizontally due to inertia. b. A person on ground (10 pts) The ball would appear to travel in a curved trajectory to someone on the ground looking at the boat. This is owing to their ability to see both the ball's vertical motion caused by gravity as well as the boat's horizontal motion.

c. Do both people see the ball hitting the same spot on the boat or different? (5 pts) The ball would strike the same point on the boat, visible to both the person on the boat and the person on the ground. This is owing to the fact that both the ball's vertical velocity caused by gravity and its horizontal motion (together with that of the boat) are constant. Because of this, regardless of the viewpoint of the person who is watching, the ball will always land in the same location on the boat Explain your trajectories for the two cases. 3. Earth and moon: a. What is the direction of gravitational force between earth and moon? (4 pts) The gravitational attraction between the earth and moon is bidirectional. As they are pulling one another. They fall together because of the attraction caused by the gravitational force between them. b. Is the Moon falling towards the earth? Explain your answer. (8 pts) Due to the gravitational pull between them—the earth has a larger pull due to its bigger mass—the moon is falling towards the earth. The moon travels in space along a gravitationally-driven orbit, which explains why it doesn't collide with the earth or move in a manner similarly to asteroids. The moon is falling freely. c. What would be the trajectory of earth’s moon if suddenly earth’s gravitation field vanished? (8 pts) It would travel in a straight line indefinitely, travelling at its current speed. Unless it was caught in another planets gravitational force. 4. Kepler’s laws: a. Kepler observed that planets sweep out equal area in equal time, but the speed at which planets move at different parts of the orbit is not constant. Why is it so? Explain your answer. (10 pts) A line from a planet to the sun sweeps equal regions in equal amounts of time, according to Kepler's Second Law, aka Law of Equal areas. The region is wider when closer but not as long and is longer when further but not as wide. But proportional in both cases. This suggests that a planet's speed as it travels around its orbit varies. When it is close to the sun, it moves more quickly; when it is farther away, it moves slower in comparison. b. Does Earth travel fastest when it’s closest to the Sun or when it’s the farthest? Explain using Kepler’s laws. (5 pts)

The gravitational pull of the sun is the cause of this speed change. The planet moves more quickly when it is nearer the sun because of the increased gravitational force. In contrast, the gravitational force is weaker and the planet moves more slowly as it gets further away from the sun. c. What kind (shape) of an orbit will allow Kepler’s area law to hold but now the planets can have constant speed throughout its orbit. Justify your choice of shape? (10 pts) The orbit would have to be circular for a planet to obey Kepler's Second Law and maintain a constant speed throughout. Because the sun and the planet are always at the same distance from each other in a circular orbit, the planet's speed and gravitational pull are also constant. 5. Summarize your understanding of what one means by “universality of a law” using Newton’s law of gravitation as an example, in no more than 100 words. If you use any resources to answer this, please cite them. (15 pts) When a law is said to be universal, it signifies that it can be applied to anything in the universe. According to Newton's law of gravity, everything in the universe is drawn to one another by an external force, which explains why some objects in the universe move in a particular way. Because it is the driving force behind the universe's objects and holds true on earth and anywhere else in the universe, it is a universal law.

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