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The space shuttle releases a satellite into a circular orbit 630 km above the Earth.
How fast must the shuttle be moving (relative to Earth's center) when the release occurs?

Respuesta :

Supernova

Answer:

7,539 m/s

Explanation:

Let's use this equation to find the gravitational acceleration of this space shuttle:

  • [tex]\displaystyle g=\frac{GM}{r^2}[/tex]

We know that G is the gravitational constant: 6.67 * 10^(-11) Nm²/kg².

M is the mass of the planet, which is Earth in this case: 5.972 * 10^24 kg.

r is the distance from the center of Earth to the space shuttle: radius of Earth (6.3781 * 10^6 m) + distance above the Earth (630 km → 630,000 m).

Plug these values into the equation:

  • [tex]\displaystyle g=\frac{(6.67\cdot 10^-^1^1 \ Nm^2kg^-^2)(5.972\cdot 10^2^4 \ kg)}{[(6.3781\cdot 10^6 \ m)+(630000 \ m)]^2}[/tex]

Remove units to make the equation easier to read.

  • [tex]\displaystyle g=\frac{(6.67\cdot 10^-^1^1 )(5.972\cdot 10^2^4 )}{[(6.3781\cdot 10^6)+(630000 )]^2}[/tex]

Multiply the numerator out.

  • [tex]\displaystyle g=\frac{(3.983324\cdot 10^1^4)}{[(6.3781\cdot 10^6)+(630000 )]^2}[/tex]

Add the terms in the denominator.

  • [tex]\displaystyle g=\frac{(3.983324\cdot 10^1^4)}{[(7008100)]^2}[/tex]

Simplify this equation.

  • [tex]\displaystyle g=8.11045189 \ \frac{m}{s^2}[/tex]

The acceleration due to gravity g = 8.11045189 m/s². Now we use the equation for acceleration for an object in circular motion which contains v and r.

  • [tex]\displaystyle a = \frac{v^2}{r}[/tex]

a = g, v is the velocity that the space shuttle should be moving (what we are trying to solve for), and r is the radius we had in the previous equation when solving for g.

Plug these values into the equation and solve for v.

  • [tex]\displaystyle 8.11045189 \ \frac{m}{s^2} = \frac{v^2}{7008100 \ m}[/tex]  

Remove units to make the equation easier to read.

  • [tex]\displaystyle 8.11045189 = \frac{v^2}{7008100}[/tex]

Multiply both sides by 7,008,100.

  • [tex]56838857.89=v^2[/tex]

Take the square root of both sides.

  • [tex]v=7539.154985[/tex]

The shuttle should be moving at a velocity of about 7,539 m/s when it is released into the circular orbit above Earth.

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