For small distances compared with the Earth’s radius, the change in gravitational acceleration can be approximated by first–order terms in h/R_E and d/R_E, where h is altitude and d is depth. Using this information, answer the following.
a. (i) Starting from the exact expression for g at altitude h, obtain its first–order approximation in terms of h/R_E.
b. (ii) Starting from the exact expression for g at depth d, obtain its first–order approximation in terms of d/R_E.
c. (iii) Hence derive an algebraic relation between h and d for which the values of g at the two locations are equal, retaining only first–order terms.
d. (iv) State the physical implication of the relation you have obtained.