Final answer:
To supply the necessary energy to remove all of Earth's mass to infinity, the variable x in the formula x/5 GM²/R is equal to 5. This represents the positive value of Earth's gravitational binding energy.
Step-by-step explanation:
The student is asking how to calculate the amount of energy required to remove all the mass of the Earth to infinity, effectively breaking its gravitational bonds. The formula provided is a fraction of Earth’s gravitational binding energy, which is given by U = -GM²/R. This formula represents the potential gravitational energy needed to be overcome to disperse the Earth's mass. The given expression in the question is a multiple of this value, x/5 GM²/R. The variable x is what the student is searching for.
To find x, we must recognize that the energy required for completely overcoming Earth's gravitational binding is equivalent to U but expressed with a proportionality constant. So, if the total energy is represented by U with the form -GM²/R, and we want to express it as a positive value for the input of energy, x would be 5 to represent the gravitational binding energy with a positive sign, meaning x = 5.