I need help with his whole problem! 5. Comets come in two varieties, long and sh

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I need help with his whole problem! 5. Comets come in two varieties, long and short period. Short period comets are those in orbits around the sun with short enough periods that astronomers have seen them return for more than one orbit, like Halley’s Comet which famously has a 76-year period. Long-period comets come unexpectedly and randomly for a single pass by the sun and then head back to great distances from the sun. Some of these could have periodic orbits, but their periods are so long that we have not observed them return (periods greater than a few hundred years). The long-period comets come from the Oort Cloud, a group of small objects orbiting the sun at many times the distance of Pluto; chaotic motions due to gravitational interactions of the objects in the cloud likely push these objects onto trajectories towards the sun. The Oort Cloud is known to be nearly spherically distributed around the sun, because long-period comets are equally likely to come from any direction. (a) Long-period comets heading towards the sun will pass through an imaginary sphere with radius equal to the Earth-Sun distance. What is the surface area of this sphere? (b) To a comet approaching this sphere, the Earth appears to be a circle. What is the area of this circle? (c) From your answers in parts (a) and (b), deduce the probability that a long-term comet will hit the Earth, if it approaches the sun from a random direction. (d) Research question: how many long period comets per year come within the Earth-Sun distance of the Sun? (e) Estimate the mean time between comet impacts with the Earth. For how long have humans lived on the Earth? For how long has life existed on the Earth? (f) A typical comet might have a radius of 500 meters and be composed of water ice, with a density of around 1 g/cm3 . For this object, what is its gravitational potential energy due to the Sun when it is the Earth’s distance away from the Sun? (g) Estimate the comet’s initial kinetic energy if it is in an orbit with a radius 10 times greater than the Pluto-Sun distance. (h) Estimate the energy the comet would have when it collides with Earth. Express in Joules and in MTons (million tons of TNT: 1MTon = 4.2×1015J). How does this energy compare to the largest nuclear bombs ever exploded? It is suspected that comet collisions could be responsible for one or more of the great extinctions seen in the fossil record of life on Earth.
I need help with his whole problem! 5. Comets come in two varieties, long and short period. Short period comets are those in orbits around the sun with short enough periods that astronomers have seen them return for more than one orbit, like Halley’s Comet which famously has a 76-year period. Long-period comets come unexpectedly and randomly for a single pass by the sun and then head back to great distances from the sun. Some of these could have periodic orbits, but their periods are so long that we have not observed them return (periods greater than a few hundred years). The long-period comets come from the Oort Cloud, a group of small objects orbiting the sun at many times the distance of Pluto; chaotic motions due to gravitational interactions of the objects in the cloud likely push these objects onto trajectories towards the sun. The Oort Cloud is known to be nearly spherically distributed around the sun, because long-period comets are equally likely to come from any direction. (a) Long-period comets heading towards the sun will pass through an imaginary sphere with radius equal to the Earth-Sun distance. What is the surface area of this sphere? (b) To a comet approaching this sphere, the Earth appears to be a circle. What is the area of this circle? (c) From your answers in parts (a) and (b), deduce the probability that a long-term comet will hit the Earth, if it approaches the sun from a random direction. (d) Research question: how many long period comets per year come within the Earth-Sun distance of the Sun? (e) Estimate the mean time between comet impacts with the Earth. For how long have humans lived on the Earth? For how long has life existed on the Earth? (f) A typical comet might have a radius of 500 meters and be composed of water ice, with a density of around 1 g/cm3 . For this object, what is its gravitational potential energy due to the Sun when it is the Earth’s distance away from the Sun? (g) Estimate the comet’s initial kinetic energy if it is in an orbit with a radius 10 times greater than the Pluto-Sun distance. (h) Estimate the energy the comet would have when it collides with Earth. Express in Joules and in MTons (million tons of TNT: 1MTon = 4.2×1015J). How does this energy compare to the largest nuclear bombs ever exploded? It is suspected that comet collisions could be responsible for one or more of the great extinctions seen in the fossil record of life on Earth.
I need help with his whole problem! 5. Comets come in two varieties, long and short period. Short period comets are those in orbits around the sun with short enough periods that astronomers have seen them return for more than one orbit, like Halley’s Comet which famously has a 76-year period. Long-period comets come unexpectedly and randomly for a single pass by the sun and then head back to great distances from the sun. Some of these could have periodic orbits, but their periods are so long that we have not observed them return (periods greater than a few hundred years). The long-period comets come from the Oort Cloud, a group of small objects orbiting the sun at many times the distance of Pluto; chaotic motions due to gravitational interactions of the objects in the cloud likely push these objects onto trajectories towards the sun. The Oort Cloud is known to be nearly spherically distributed around the sun, because long-period comets are equally likely to come from any direction. (a) Long-period comets heading towards the sun will pass through an imaginary sphere with radius equal to the Earth-Sun distance. What is the surface area of this sphere? (b) To a comet approaching this sphere, the Earth appears to be a circle. What is the area of this circle? (c) From your answers in parts (a) and (b), deduce the probability that a long-term comet will hit the Earth, if it approaches the sun from a random direction. (d) Research question: how many long period comets per year come within the Earth-Sun distance of the Sun? (e) Estimate the mean time between comet impacts with the Earth. For how long have humans lived on the Earth? For how long has life existed on the Earth? (f) A typical comet might have a radius of 500 meters and be composed of water ice, with a density of around 1 g/cm3 . For this object, what is its gravitational potential energy due to the Sun when it is the Earth’s distance away from the Sun? (g) Estimate the comet’s initial kinetic energy if it is in an orbit with a radius 10 times greater than the Pluto-Sun distance. (h) Estimate the energy the comet would have when it collides with Earth. Express in Joules and in MTons (million tons of TNT: 1MTon = 4.2×1015J). How does this energy compare to the largest nuclear bombs ever exploded? It is suspected that comet collisions could be responsible for one or more of the great extinctions seen in the fossil record of life on Earth.

Explanation / Answer

a. Since the distance between earth and sun or the orbital radius of earth=1.496*10^11m

We know that the surface area of sphere=4/3piR^3

=4/3*3.14*(1.496*10^11)^3

=4/3*3.14*3.34*10^33

=41.9504*10^33/3

=13.9*10^33m

=14*10^33m

b. The radius of earth=6371 km

Area of the circle=pi*R2

where R= radius of the earth

Area=3.14*(6371)^2

=3.14* 40589641

=127,451,472.7km2

e. The humans have lived on the earth for about 2.2 million years ago. Life started on earth about 3.5 billion years ago from the most primitive unicellular microbes and evolved into complex animals.

f. Gravitational potential energy U=-GMm/r

where r= radius of the body=500m

M=mass of sun=1.989*10^30kg

m=mass of comet=?

First we need to find the mass of comet

Since mass=density/volume

density of comet=1g/cm3=1000kg/m3

volume of comet=4/3pi R3

=4/3*3.14*(500)^2

=4/3*3.14*250,000m3

=3,140,000/3

=1,0466,666.66m3

Mass=density/volume

=1000kg/m3/10466,666.66m3

=9.5*10^-5kg

Now gravitational potential energy due to sun=-GMm/r

G= gravitational constant=6.67*10^-11m3kg-1s-2

M=1.989*10^30kg

m=9.5*10^-5kg

r=500m

U=-6.67*10^-11*1.989*10^30*9.5*10^-5/500

=126.03*10^14/500

=0.25kgm2s-2

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