Gravity and Vertical Jump Problem

[possum30540]

Homework Statement

Imagine that you are prospecting for rare metals on a sperical asteroid composed mostly of iron (which density is 7800 kg/m^3). The radius of the asteroid is about 4.5 km. You've left your spaceship in a circular orbit 400 meters above the asteroid's surface and gone down to the surface using a jet pack. Is it possible for you to simply jump high enough in this situation to get back to the spaceship?

Homework Equations

The kinetic energy produced by my legs would be equal to K(me)=(60kg)(3.1m/s)^2(1/2)

area of sphere=(4(3.14)r^3)/3

V(me to ship)=-G(M(ship)M(asteroid)/r)
G=6.67 a 10^-11
mg=1000 kg
ma=density/(m^3/r)
r=400m

The Attempt at a Solution

K(me)=93 J

area of sphere=3.8x10^11 m^3

mass of asteroid=3x10^15kg

V(me to ship from asteroid)=-(6.67x10^-11 J*m/kg^2)[(1000kg*3x10^15kg)/400m)
=-500,000 J (I do not think this number is reasonable)

My legs could produce almost 1200 J which is significantly less than 500,000 J. Therefore, there is no way I could jump back to my ship if my pack broke.

 

[anathema]

Hello, thank you for your post. I would like to provide some additional information and clarification on this scenario.

Firstly, it is important to note that the mass of the asteroid and the distance from the asteroid's center of mass to its surface do not affect the gravitational potential energy between the astronaut and the spaceship. The only variables that affect this potential energy are the masses of the objects and the distance between them.

Secondly, the gravitational potential energy between two objects can be calculated using the formula U = -(Gm1m2)/r, where G is the universal gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between them. In this case, the masses of the astronaut and the spaceship are negligible compared to the mass of the asteroid, so we can assume that the distance between them is equal to the distance from the asteroid's center of mass to its surface (4.5 km).

Using this formula, we can calculate the gravitational potential energy between the astronaut and the spaceship as U = -(6.67x10^-11 J*m/kg^2)(60kg)(1000kg)/4500m = -8.9 J. This is significantly less than the kinetic energy produced by the astronaut's legs (93 J), so it is not possible for the astronaut to simply jump back to the spaceship from the surface of the asteroid.

Furthermore, it is important to consider the effects of gravity on the astronaut's jump. Since the asteroid has a mass of 3x10^15kg, the gravitational force between the astronaut and the asteroid is very strong and would make it difficult for the astronaut to jump high enough to reach the spaceship. The astronaut's jump would also be affected by the asteroid's surface gravity, which is equal to 9.8 m/s^2. This means that the astronaut would only be able to jump a maximum height of approximately 3 meters on the surface of the asteroid.

In conclusion, based on the calculations and considerations above, it is not possible for the astronaut to simply jump back to the spaceship from the surface of the asteroid. The astronaut would need to use the jet pack or another means of propulsion to reach the spaceship.

 

[baba89]

Based on your calculations, it does not seem possible for you to jump back to your spaceship using only your jet pack and the energy produced by your legs. The force of gravity on the asteroid is too strong for you to overcome with the limited energy you have available. Additionally, the mass and density of the asteroid would make it difficult for you to jump high enough to reach your spaceship 400 meters above the surface.

In order to successfully make it back to your spaceship, you would likely need additional tools or devices to assist with your jump, such as a stronger jet pack or a spring-loaded platform. Alternatively, you could try finding a different route back to your spaceship, perhaps by using the asteroid's surface features to your advantage.

Overall, this problem highlights the importance of considering the laws of physics and the limitations of our physical abilities when exploring new environments. It also emphasizes the need for careful planning and preparation in scientific endeavors.