The escape velocity on Planet X is the speed the rocket needs in order to never fall back down again in a universe in which X is the only object. In order for the rocket to not fall back down, its velocity to never become negative, how far away does the rocket have to be when the velocity is 0?
To determine the distance at which the rocket's velocity becomes 0, we first need to understand the concept of escape velocity and its relation to gravitational potential energy.
Escape velocity is the minimum velocity an object needs to escape the gravitational pull of a celestial body. In this case, we consider Planet X to be the only object in the universe.
The formula for escape velocity (Ve) can be derived from the equation for gravitational potential energy:
Ve = √(2GM/r)
Where:
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2)
- M is the mass of the planet or celestial body (in this case, the mass of Planet X)
- r is the distance from the center of the planet to the rocket's starting point (distance from the center of Planet X to the rocket)
To find the distance at which the rocket's velocity becomes zero (r0), we set the escape velocity equation equal to zero and solve for r:
0 = √(2GM/r0)
Squaring both sides of the equation:
0 = 2GM/r0
Next, we isolate r0 by rearranging the equation:
r0 = 2GM/0
Since dividing by zero is undefined, there is no distance at which the rocket's velocity becomes zero. In mathematical terms, the rocket would never reach zero velocity if it continues traveling away from Planet X.
Thus, the rocket would need to be infinitely far away from Planet X for its velocity to never become negative.