What is the Kinetic Energy and Velocity of a Falling Asteroid?

In summary: Then, it all becomes kinetic energy."In summary, to solve this problem, one must use the formula for gravitational potential energy between two bodies, which is U = -GMm/r, where G is the universal gravitational constant. By setting the potential energy at a large distance to zero, one can use the principle of conservation of energy to calculate the kinetic energy of the asteroid as it falls to Earth. This can then be used to find the minimum speed at which it strikes the Earth's surface.
  • #1
clope023
992
131
[SOLVED] Kinetic Energy and Velocity

Homework Statement



The gravitational pull of the Earth on an object is inversely proportional to the square of the distance of the object from the center of the earth. At the Earth's surface this force is equal to the object's normal weight mg, where g=9.8, and at large distances, the force is zero.

a) If a 50000kg asteroid falls to Earth from a very great distance away, how much kinetic energy will it impart to our planet? You can ignore the effects of the Earth's atmosphere.
Express your answer using two significant figures.

b) What will be its minimum speed as it strikes the Earth's surface?


Homework Equations



Wtot = 1/2m(v2)^2-1/2m(v1)^2

Wgrav = mg


The Attempt at a Solution



a) W = mg = (50000kg)(9.8m/s^2) = 4.9 x 10^5 J (wrong)

b) W = 1/2mv^2, v = [tex]\sqrt{2W/m}[/tex] = 4.4 (definetly wrong)

as you can see I tried using the work energy theorem with the given data, but nothing was right, any help is appreciated.
 
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  • #2
W = mg is a formula for the weight of something near the Earth's surface (not work).

Hint: Look up the gravitational PE between two bodies as a function of distance.
 
  • #3
Doc Al said:
W = mg is a formula for the weight of something near the Earth's surface (not work).

Hint: Look up the gravitational PE between two bodies as a function of distance.

but no distance was given, I know Ugrav = mgd, d being distance...
 
  • #4
clope023 said:
but no distance was given, I know Ugrav = mgd, d being distance...
Ug = mgd only applies near the Earth's surface, which is not the case here. Look for another version of a gravitational PE formula.
 
  • #5
Doc Al said:
Ug = mgd only applies near the Earth's surface, which is not the case here. Look for another version of a gravitational PE formula.

so I guess you're talking about the gravitational PE formula where the force approaches zero:

U = -GMm/r

but no distance is given so I can't find a radius, I'm assuming you're using:

Ugrav + Wtot = K

but I don't understand how I can get it without a given distance so I can find a radius for the potential energy, do I use the radius of the earth?

edit: so apparently -GMm/r = -1/2mv^2, but again I'm stumped about the radius.
 
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  • #6
tried it with the Earth's radius and got the answer for part a)

GMM/r_earth = K = 3.1x10^12 J w00t!

now for part b)

would it be [tex]\sqrt{2K/m}[/tex]?
 
  • #7
lol, I'm answering all my own questions.

it is what I just posted above, thanks to Doc Al for pointing me in the right direction with the proper formula.
 
  • #8
Good work!
 
  • #9
can anyone summarize how to solve this problem, i do not understand how to use this formula GMm/r

what is G,? 9.8?, does it give you potential energy of the object? doesn't the question ask for kinetic?

thanks for the help
 
  • #10
G is the universal gravitational constant, which appears in Newton's law of gravity. Since mechanical energy is conserved, knowing the change in gravitational potential energy allows you to calculate the change in kinetic energy. Read: http://hyperphysics.phy-astr.gsu.edu/Hbase/gpot.html#ufm"
 
Last edited by a moderator:
  • #11
Use conservation of energy. At very large distances, the asteroid has zero potential energy. We can see this is true because U = k/r (k is a constant), so as r tends to infinity, U tends to zero, so very large r we can neglect U (gravitational energy). Then it becomes trivial to solve using conservation of energy. It starts with at least zero kinetic energy.
 

Related to What is the Kinetic Energy and Velocity of a Falling Asteroid?

1. What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion. It is calculated by multiplying the mass of the object by the square of its velocity, and is measured in joules.

2. How does kinetic energy relate to velocity?

Kinetic energy is directly proportional to velocity. This means that as an object's velocity increases, its kinetic energy also increases. The relationship between kinetic energy and velocity is expressed by the equation KE = 1/2 * mv^2, where m is the mass of the object and v is its velocity.

3. Can kinetic energy be negative?

No, kinetic energy cannot be negative. Since velocity is squared in the equation for kinetic energy, it will always result in a positive value. Negative velocity simply indicates a change in direction, not a decrease in energy.

4. How is kinetic energy different from potential energy?

Kinetic energy is the energy an object possesses due to its motion, while potential energy is the energy an object possesses due to its position or state. Kinetic energy can be converted into potential energy and vice versa, but they are two distinct forms of energy.

5. How does mass affect kinetic energy?

Mass has a direct relationship with kinetic energy. This means that as an object's mass increases, its kinetic energy also increases. This can be seen in the equation for kinetic energy, as mass is directly multiplied by velocity squared.

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