Dying to get away?
When you jump up into the air you'll come back down to the Earth with a bump. That's not because the laws of nature categorically forbid you leaving the Earth, but because your jump isn't powerful enough to escape the Earth's gravitational field. To do that, the speed of your jump would have to exceed, or be equal to, the Earth's escape velocity which we can calculate quite easily as follows.
When you jump into the air your kinetic energy is
where is your mass and is your velocity. The potential energy you experience due to the Earth’s gravitational pull is
where is again your mass, is the mass of the Earth and is the Earth’s radius. For you to be able to escape from Earth, we need
Solving for velocity gives
The Earth’s escape velocity, , is defined to be the smallest velocity that allows an object to escape, so
Filling in the values
Note that our formula for escape velocity is independent on the mass of the object that is trying to escape, as cancels out. So in theory you would need to achieve the same velocity to escape Earth as, say, an elephant. We should point out, however, that our calculation ignores the effect of air resistance which would effect you and the elephant differently. What is more, if you reached such a high velocity within the Earth's atmosphere, you would burn up. To avoid this, you or the elephant should first get yourself into an orbit in which the Earth's atmosphere is weak or non-existent, and then accelerate to the escape velocity needed to escape from that orbit. Find out more here.
11182 m/s is about 40255 km/h
Could you check that 11182 m/s "translates" to 671 km/h ?? The units for G should be m˄3/kg s˄2
The gravitational constant should have units m^3/kg/s^2 not m^3/kg/s. This will asure that v_earth will have a unit of m/s. To transform this unit to km/h you have to multiply by 3.6. This gives us a speed of 40'255 km/h. I cant't figure out where the 671 km/h comes from.