Question

What work must be done to place a satelite of mass 30kg in orbit at a height of 50km above the earth with a velocity of 800m sec?

Answer 1

Approximately
`2.4 * 10^(7)\; {\rm J}`.

Step-by-step explanation:

When an object of mass
`m` is moving at a speed of
`v`, the kinetic energy of that object will be
`(1/2)\, m\, v^(2)`.

Kinetic energy of this satellite when in orbit:

`\begin{aligned}\text{KE} &= (1)/(2)\, m\, v^(2) \\ &= (1)/(2)* 30\; {\rm kg} * (800\; {\rm m\cdot s^(-1)}) \\ &\approx 9.60 * 10^(6)\; {\rm J} \end{aligned}`.

The gravitational field near the surface of the earth is approximately uniform:
`g \approx 9.81\; {\rm m\cdot s^(-2)}`. In a uniform gravitational field, raising an object of mass
`m` by a height of
`h` will require
`m\, g\, h` of work.

Apply unit conversion and ensure that the height of the satellite is measured in standard units (meters):

`h = 50\; {\rm km} = 50000\; {\rm m}`.

Work required to bring this satellite from a ground to a height of
`50000\; {\rm m}`:

`\begin{aligned}m\, g\, h &= 30\; {\rm kg} * 9.81\; {\rm m\cdot s^(-2)} * 50000\; {\rm m} \\ &\approx 1.47 * 10^(7)\; {\rm J}\end{aligned}`.

Work required in total will be approximately:

`9.60 * 10^(6)\; {\rm J} + 1.47* 10^(7)\: {\rm J} \approx 2.4 * 10^(7)\; {\rm J}`.