Darren the astronaut traveled to Planet U-427, a planet that is as big as Earth but is half the mass of Earth. What can you say about Darren's weight in this planet?
With half the mass, he would weigh half as much as on earth.
Darren's weight on the planet will be half of the weight as in earth as the weight on the planet will be
W = (mg)/2
Since Planet U-427 only has half the mass of Earth, you can infer that the gravitational force is less on this planet. Thus, Darren's weight on this planet is less than his weight on Earth.
To determine Darren's weight on Planet U-427, we can use the equation for gravitational force:
F = (G * m1 * m2) / r^2
Where:
- F is the gravitational force between two objects,
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
- m1 and m2 are the masses of the two objects, and
- r is the distance between the centers of the two objects.
In this case, m1 represents the mass of Darren, and m2 represents the mass of Planet U-427.
Given that Planet U-427 is as big as Earth but has half the mass, we can assume that Darren's mass remains the same since it is independent of the planet. Therefore, Darren's mass (m1) would be the same as his mass on Earth.
Now, to determine Darren's weight on Planet U-427, we need to compare the gravitational force on him on Planet U-427 to the gravitational force on him on Earth. The gravitational force on a person's weight on Earth is given by:
F-Earth = (G * m1 * m2-Earth) / r-Earth^2
Since Darren's mass (m1) remains the same, we can compare the weights by using the ratio of the gravitational forces on Planet U-427 (F-U427) to Earth (F-Earth):
F-U427 / F-Earth = (G * m2-U427) / (G * m2-Earth)
This ratio can be simplified as follows:
F-U427 / F-Earth = m2-U427 / m2-Earth
Since Planet U-427 has half the mass of Earth (m2-U427 = 0.5 * m2-Earth):
F-U427 / F-Earth = 0.5 * m2-Earth / m2-Earth = 0.5
Therefore, Darren's weight on Planet U-427 would be half of his weight on Earth.