In the distant future, an astronaut is exploring a newly discovered planet which has almost exactly the same mass as Earth, but has a diameter almost exactly half of that of Earth. On Earth, the astronaut weighed 1,390 N (with their full spacesuit and gear). What would their weight be on this new planet? Explain your answer to gain full marks.

To find the weight of the astronaut on the new planet, we can use the formula for gravitational force:

F = (G * m1 * m2) / r^2

Where: F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.

Since the mass of the astronaut remains the same, we can ignore it in this calculation. The only variables that change are the gravitational constant and the distance between the centers of the two objects.

Given that the new planet has almost exactly the same mass as Earth, we can assume that the gravitational constant remains the same. However, the distance between the centers of the two objects is different.

The diameter of the new planet is almost exactly half of that of Earth. Since the diameter is the distance across the center of a sphere, we can assume that the distance between the centers of the two objects is also half of that on Earth.

Therefore, the distance between the centers of the two objects on the new planet is half of the distance on Earth. This means that the distance squared is also half of the distance squared on Earth.

Now, let's substitute these values into the formula:

F_new = (G * m1 * m2) / (r_new)^2

Since the mass of the astronaut remains the same, we can ignore it:

F_new = (G * m1) / (r_new)^2

Since the gravitational constant and the mass of the astronaut are constants, we can simplify the equation to:

F_new = k / (r_new)^2

Where k is a constant.

Since the distance between the centers of the two objects on the new planet is half of that on Earth, we can substitute this value into the equation:

F_new = k / (0.5 * r)^2

Simplifying further:

F_new = k / (0.25 * r^2)

Since k is a constant, we can ignore it for the purpose of this calculation.

Therefore, the weight of the astronaut on the new planet would be four times their weight on Earth.

In this case, the astronaut weighed 1,390 N on Earth. Therefore, their weight on the new planet would be:

Weight_new = 4 * Weight_earth Weight_new = 4 * 1,390 N Weight_new = 5,560 N

So, the weight of the astronaut on the new planet would be 5,560 N.