The magnitude of the gravitational attraction between two particles of massm1 and m2 is given by m.m. \left|F_{\mathrm{grav}}(r)\right|=G \frac{m_{1} m_{2}}{r^{2}} where r is the distance between the two particles and

the force that each particle exerts on the other is directed along the radial direction between the two particles, and G is the universal gravitational constant. \text { (a) What is the potential energy function } U=U_{a}(\tau) \text {, if we take } U_{0}=\lim U_{a}(\tau)=0 \text { ? } \text { That is, we take our reference point out at } r=\infty \text {. } (b) How much work W, must be done against the conservative gravitational force toincrease the separation distance from r = R to r= R+h? (d) We are now going to repeat the calculation in part (c) except this time we are going to approximate the force of gravity as a constant force F grav = -mg, where g = is the constant magnitude of gravity that is assumed to be independent of the separation distance between the two attracting masses. This is nothing new, when we apply New-ton's second law to trajectory problems involving gravity on the earth's surface, such as computing the path of a cannon ball launched from the ground, we take the force of gravity as a constant.GMe Compute the work W, done in moving the same object of mass m in part c from the ground at y=0 height y=h against the constant force of gravity f grav -mg (e) We now want to compare the exact result for the work found in part (c) with the approximate value of work found in part (d). What is the size of the relative error between the work found in part (c) and (d)? Hint: W.L.O.G. we can take W,reference measure. Show that the relative error is \frac{W_{g}-W_{\text {grav }}}{W_{\text {grav }}}=\mathcal{O}\left(\frac{h}{R_{e}}\right) (f) The mean radius of the earth is approximately 6370 km. Use the relative error thatyou find to predict how high you can go above the earth's surface and safely ignore thefact that gravity is changing with height when you compute the work done in lifting anobject from the earth's surface.