Motion of Orbiting Satellites Practice

Sample questions

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1. Sample 1difficulty 4/5

   How much energy must be added to move a satellite of mass $m$ from a circular orbit of radius $r$ to a circular orbit of radius $2r$ around a planet of mass $M$?

   • A

     $-\dfrac{GMm}{4r}$

   • B

     $+\dfrac{GMm}{r}$

   • C

     $+\dfrac{GMm}{4r}$

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   • D

     $+\dfrac{GMm}{2r}$

   Why

   $E_2 - E_1 = -GMm/(4r) - (-GMm/(2r)) = GMm/(4r)$.

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2. Sample 2difficulty 4/5

   A satellite of mass $m$ orbits at radius $r$ around a planet of mass $M$. Its orbital kinetic energy is:

   • A

     $\dfrac{2GMm}{r}$

   • B

     $-\dfrac{GMm}{2r}$

   • C

     $\dfrac{GMm}{r}$

   • D

     $\dfrac{GMm}{2r}$

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   Why

   From $mv^2/r = GMm/r^2$, so $K = \tfrac12 mv^2 = GMm/(2r)$.

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3. Sample 3difficulty 4/5

   Planet X has twice the mass and half the radius of Earth. What is the ratio of escape velocity from X to escape velocity from Earth, $v_X/v_E$?

   • A

     $\sqrt{2}$

   • B

     $2$

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   • C

     $4$

   • D

     $\tfrac{1}{2}$

   Why

   $v_{esc} = \sqrt{2GM/R}$. Ratio is $\sqrt{(2M)/(R/2)} / \sqrt{M/R} = \sqrt{4} = 2$.

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4. Sample 4difficulty 4/5

   A moon at orbital radius $r$ has period $T$. A second moon orbits the same planet at radius $9r$. What is its period in terms of $T$?

   • A

     $81T$

   • B

     $27T$

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   • C

     $3T$

   • D

     $9T$

   Why

   Kepler's third law: $T^2 \propto r^3$. So $T_2/T = (9)^{3/2} = 27$.

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5. Sample 5difficulty 4/5

   A satellite has period $T = 24$ h around Earth at radius $R_g$. A second satellite has period $T = 3$ h. What is its orbital radius?

   • A

     $R_g/4$

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   • B

     $R_g/3$

   • C

     $R_g/8$

   • D

     $R_g/2$

   Why

   $T^2 \propto r^3$. $r_2/R_g = (3/24)^{2/3} = (1/8)^{2/3} = 1/4$.

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