Worksheet: Mechanical Energy Conservation in Orbits

In this worksheet, we will practice calculating the kinetic energy required to maintain circular orbit around a star or planet.

Q1:

The figure shows a planet orbiting a star along an elliptical path.

At which of the labeled points on the figure would the gravitational potential energy of the planet have the highest value?

  • APoint A
  • BPoint C
  • CPoint E
  • DPoint D
  • EPoint B

At which of the labeled points on the figure would the kinetic energy of the planet be greatest?

  • APoint E
  • BPoint D
  • CPoint B
  • DPoint A
  • EPoint C

As the planet moves from point D to point C, does its gravitational potential energy increase, decrease, or stay the same?

  • AIt stays the same.
  • BIt decreases.
  • CIt increases.

As the planet moves from point C to point B, does its kinetic energy increase, decrease, or stay the same?

  • AIt stays the same.
  • BIt decreases.
  • CIt increases.

Q2:

The diagram shows two planets orbiting a star. Both planets have the same mass. Planet 2 orbits the star at a radius twice that of planet 1.

What is the ratio of the kinetic energy of planet 2 to that of planet 1?

  • A2
  • B14
  • C4
  • D12
  • E1

Q3:

Which of the five lines on the graph correctly shows how the kinetic energy of an object in a circular orbit varies with the radius of the orbit?

  • AThe black line
  • BThe red line
  • CThe blue line
  • DThe green line
  • EThe purple line

Q4:

The diagram shows two planets orbiting a star. Both planets have the same kinetic energy. Planet 2 orbits the star at a radius three times that of planet 1.

What is the ratio of the mass of planet 2 to that of planet 1?

Q5:

At a distance 𝑅 away from a planet, an object with a mass 𝑀 has a gravitational potential energy of 4.00 GJ. Its velocity is tangential to the line connecting the center of mass of the object and the planet.

If the object has a kinetic energy of 2.00 GJ, what will happen to it?

  • AIt will move closer to the planet, gaining kinetic energy. Its increased kinetic energy will cause it to move further away from the planet again, and the object will follow an elliptical orbit.
  • BIt will orbit the planet along a circular path.
  • CIt will completely escape the planet’s gravitational pull.
  • DIt will move further away from the planet before being pulled back toward it by gravity, following an elliptical orbit.

If the object has a kinetic energy of 3.00 GJ, what will happen to it?

  • AIt will move further away from the planet before being pulled back toward it by gravity, following an elliptical orbit.
  • BIt will move closer to the planet, gaining kinetic energy. Its increased kinetic energy will cause it to move further away from the planet again, and the object will follow an elliptical orbit.
  • CIt will orbit the planet along a circular path.
  • DIt will completely escape the planet’s gravitational pull.

Q6:

Which of the following formulas gives the ratio of the kinetic energy, 𝐸K, to the gravitational potential energy, 𝐸P, of an object that is in a circular orbit?

  • A𝐸=βˆ’πΈ4KP
  • B𝐸=βˆ’πΈ2KP
  • C𝐸=βˆ’2𝐸KP
  • D𝐸=βˆ’πΈKP
  • E𝐸=βˆ’4𝐸KP

Q7:

The 2001 Mars Odyssey is a spacecraft orbiting Mars. The radius of its orbit is 3,790 km, and it has a mass of 376 kg. What is the kinetic energy of the spacecraft? Use a value of 6.42Γ—10 kg for the mass of Mars and 6.67Γ—10 m3/kgβ‹…s2 for the universal gravitational constant. Give your answer to 3 significant figures.

Q8:

Which of the following formulas correctly describes the relation between the kinetic energy of an object in a circular orbit and the radius of the orbit?

  • A𝐸∝1π‘ŸοŒͺ
  • BπΈβˆπ‘ŸοŒͺ
  • CπΈβˆβˆšπ‘ŸοŒͺ
  • D𝐸∝1π‘ŸοŒͺ
  • EπΈβˆπ‘ŸοŒͺ

Q9:

An object with a mass of 200 kg is orbiting Earth. Its orbit is circular with a radius of 40,000 km.

What is the gravitational potential energy between the object and Earth? Use a value of 5.97Γ—10οŠͺ kg for the mass of Earth and a value of 6.67Γ—10 m3/kgβ‹…s2 for the universal gravitational constant. Give your answer to 3 significant figures.

What is the kinetic energy of the object? Give your answer to 3 significant figures.

If the object was at a radial distance of 30,000 km away from Earth instead, what would the gravitational potential energy between the object and Earth be? Give your answer to 3 significant figures.

If the object was to move from a position in its orbit 40,000 km away from Earth to a position 30,000 km away from Earth and all of the gravitational potential energy lost by the object and Earth’s system was gained by the object as kinetic energy, what would the kinetic energy of the object be? Give your answer to 3 significant figures.

How much greater is the kinetic energy of the object than the kinetic energy needed to sustain circular orbit at a distance of 30,000 km away from Earth? Give your answer to 3 significant figures.

Q10:

A spacecraft that has been launched into space is moving along a circular path around Earth. The radius of the orbit is 7,000 km, and the spacecraft has a mass of 2,200 kg. What is the kinetic energy of the spacecraft? Use a value of 5.97Γ—10οŠͺ kg for the mass of Earth and 6.67Γ—10 m3/kgβ‹…s2 for the universal gravitational constant. Give your answer to 3 significant figures.

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