An astronaut exploring a distant solar system lands on an unnamed planet with radius 3,860 km. When the astronaut jumps upward with an initial speed of 3.5 m/s, she rises to a height of 0.62 m. What is the mass of the planet to within 0.1%?

To determine the mass of the planet, we can use the concept of gravitational potential energy.

The potential energy of the astronaut at the highest point of her jump can be given as:

PE = m * g * h

Where:
PE is the potential energy
m is the mass of the astronaut
g is the acceleration due to gravity
h is the height

At the highest point, the potential energy is equal to the initial kinetic energy of the astronaut:

PE = KE

Since the initial speed of the astronaut is given as 3.5 m/s, the kinetic energy can be calculated as:

KE = (1/2) * m * v^2

Where:
KE is the kinetic energy
m is the mass of the astronaut
v is the initial velocity

Setting the potential energy and kinetic energy equal to each other, we have:

m * g * h = (1/2) * m * v^2

Simplifying the equation by canceling out m, we get:

g * h = (1/2) * v^2

Now, we need to find the acceleration due to gravity, g, on this unnamed planet. The acceleration due to gravity can be calculated using Newton's law of universal gravitation:

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

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

In this case, the gravitational force is equal to the weight of the astronaut, which can be given as:

F = m * g

Setting the two equations equal to each other, we have:

m * g = G * (m * M) / R^2

Where:
M is the mass of the unnamed planet
R is the radius of the unnamed planet

Simplifying the equation by canceling out m, we get:

g = (G * M) / R^2

Now, we can substitute this value of g in the equation derived earlier:

[(G * M) / R^2] * h = (1/2) * v^2

Simplifying further, we get:

M = (v^2 * R^2) / (2 * G * h)

Given that:
v = 3.5 m/s
R = 3,860 km = 3,860,000 m
h = 0.62 m
G = 6.67 × 10^-11 N(m/kg)^2

Plugging in these values, we can calculate the mass of the planet:

M = (3.5^2 * (3,860,000)^2) / (2 * (6.67 × 10^-11) * 0.62)

M ≈ 5.938 × 10^24 kg

Therefore, the mass of the planet is approximately 5.938 × 10^24 kg, to within 0.1%.

To find the mass of the planet, we can use the law of universal gravitation, which states:

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

Where:
F is the gravitational force between two objects,
G is the gravitational constant (approximately 6.674 * 10^-11 N(m/kg)^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, we can consider the astronaut as the first object and the planet as the second object. Since the astronaut is jumping, we assume that the force exerted on the astronaut by the planet is equal to the gravitational force between them. Therefore, we can write:

F = m1 * g

Where:
m1 is the mass of the astronaut, and
g is the acceleration due to gravity on the planet.

Now, let's calculate the acceleration due to gravity on the planet:

g = G * m2 / r^2

Rearranging the equation:

m2 = g * r^2 / G

To find the mass of the planet, we need to calculate g first. Given that the astronaut rises to a height of 0.62 m and has an initial speed of 3.5 m/s, we can use the following equation of motion:

v^2 = u^2 + 2 * a * s

Where:
v is the final velocity (0 m/s since the astronaut reaches the highest point),
u is the initial velocity (3.5 m/s),
a is the acceleration (the acceleration due to gravity, g),
and s is the displacement (0.62 m).

Rearranging the equation:

a = (v^2 - u^2) / (2 * s)

Now we can calculate the acceleration due to gravity:

g = (0 - (3.5 m/s)^2) / (2 * 0.62 m)

Once we have g, we can substitute it back into the equation to find the mass of the planet:

m2 = g * (3,860 km)^2 / G

Note: 3,860 km should be converted to meters (3,860 km = 3,860,000 m) before substituting it into the equation.

Finally, by solving the equation, we can find the mass of the planet to within 0.1%.

First find g for the planet

g = v^2/2y
Now we can use universal gravitation to find M
M = r^2 g/G