
Spacecraft designers have decided that a 400.0−m−radius
 spacecraft only needs to produce an acceleration of 0.500 g. What speed would produce that acceleration?

To determine the speed that would produce an acceleration of \(0.500 \, g\) (where \(g = 9.8 \, \text{m/s}^2\)) for a spacecraft with a radius of 400 meters, we can use the formula for centripetal acceleration:

\[
a_c = \frac{v^2}{r}
\]

Where:
- \(a_c\) is the centripetal acceleration,
- \(v\) is the speed,
- \(r\) is the radius of the spacecraft.

We are given:
- \(a_c = 0.500 \, g = 0.500 \times 9.8 \, \text{m/s}^2 = 4.9 \, \text{m/s}^2\),
- \(r = 400 \, \text{m}\).

Now, solve for \(v\):

\[
v = \sqrt{a_c \times r}
\]

Substituting the values:

\[
v = \sqrt{4.9 \, \text{m/s}^2 \times 400 \, \text{m}} = \sqrt{1960} \, \text{m/s}
\]

\[
v \approx 44.3 \, \text{m/s}
\]

Thus, the speed required is approximately \(44.3 \, \text{m/s}\).

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Would you like further details or have any other questions? Here are some related questions to explore:

1. How does centripetal acceleration vary with radius and velocity?
2. What would happen if the radius of the spacecraft increased?
3. How do different levels of artificial gravity affect long-term human health?
4. What are other ways to generate artificial gravity in spacecraft?
5. How does Earth's gravity compare to the 0.5 g of acceleration in space?

**Tip:** Always double-check the units when dealing with acceleration and velocity to ensure consistency.

Spacecraft designers have decided that a 400.0−m−radius spacecraft only needs to produce an acceleration of 0.500 g. What speed would produce that acceleration?

Discover how to calculate the speed required to produce a centripetal acceleration of 0.500 g for a spacecraft with a radius of 400 meters. This physics problem involves key concepts such as centripetal acceleration and the formula v = sqrt(a_c * r), making it suitable for students in Grades 11-12.

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