Questionon the Moon? On Earth? Suppose the mass of a fully loaded module in which astronauts take off from the Moon is $1.00 \times 10^{4} \mathrm{~kg}$ . The thrust of its engines is $3.00 \times 10^{4} \mathrm{~N}$ . (a) Calculate the module's magnitude of acceleration in a vertical takeoff from the Moon. (b) Could it lift off from Earth? If not, why not? If it could, calculate the magnitude of its acceleration.

Studdy Solution

STEP 1

1. The mass of the module is $1.00 \times 10^{4} \, \mathrm{kg}$ .
2. The thrust of the engines is $3.00 \times 10^{4} \, \mathrm{N}$ .
3. The acceleration due to gravity on the Moon is $1.625 \, \mathrm{m/s^2}$ .
4. The acceleration due to gravity on Earth is $9.81 \, \mathrm{m/s^2}$ .

STEP 2

1. Calculate the module's magnitude of acceleration during a vertical takeoff from the Moon.
2. Determine if the module could lift off from Earth and calculate the magnitude of its acceleration if possible.

STEP 3

Calculate the module's magnitude of acceleration on the Moon.
The net force acting on the module is the thrust minus the gravitational force on the Moon. The gravitational force on the Moon is calculated as:
$F_{\text{gravity, Moon}} = m \times g_{\text{Moon}}$
where $m = 1.00 \times 10^{4} \, \mathrm{kg}$ and $g_{\text{Moon}} = 1.625 \, \mathrm{m/s^2}$ .
$F_{\text{gravity, Moon}} = 1.00 \times 10^{4} \times 1.625 = 1.625 \times 10^{4} \, \mathrm{N}$
The net force $F_{\text{net, Moon}}$ is:
$F_{\text{net, Moon}} = F_{\text{thrust}} - F_{\text{gravity, Moon}}$ $F_{\text{net, Moon}} = 3.00 \times 10^{4} - 1.625 \times 10^{4} = 1.375 \times 10^{4} \, \mathrm{N}$
Using Newton's second law, $F = ma$ , the acceleration $a_{\text{Moon}}$ is:
$a_{\text{Moon}} = \frac{F_{\text{net, Moon}}}{m}$ $a_{\text{Moon}} = \frac{1.375 \times 10^{4}}{1.00 \times 10^{4}} = 1.375 \, \mathrm{m/s^2}$

STEP 4

Determine if the module could lift off from Earth and calculate the magnitude of its acceleration if possible.
First, calculate the gravitational force on Earth:
$F_{\text{gravity, Earth}} = m \times g_{\text{Earth}}$ $F_{\text{gravity, Earth}} = 1.00 \times 10^{4} \times 9.81 = 9.81 \times 10^{4} \, \mathrm{N}$
Compare the thrust to the gravitational force on Earth:
Since $F_{\text{thrust}} = 3.00 \times 10^{4} \, \mathrm{N}$ and $F_{\text{gravity, Earth}} = 9.81 \times 10^{4} \, \mathrm{N}$ , the thrust is less than the gravitational force. Therefore, the module cannot lift off from Earth.
The module's magnitude of acceleration during a vertical takeoff from the Moon is $1.375 \, \mathrm{m/s^2}$ , and it cannot lift off from Earth due to insufficient thrust.

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