A rock is launched vertically upward from sea level with an initial speed of 500\,m\,s−1. Neglect air resistance in all calculations.
(a) Assuming the gravitational field is uniform with magnitude g0=9.80ms−2, calculate the maximum height hc reached by the rock above sea level.
(b) Treating the Earth–rock system with universal gravitation, derive an expression for the maximum radial distance rmax from Earth’s center in terms of the given quantities. Then determine the more accurate maximum height hv=rmax−RE.
(c) Compute the percent error introduced by using the uniform–g result from part (a) instead of the value from part (b).
(d) Discuss how the magnitude of this error would change if the initial launch speed were doubled, supporting your explanation with qualitative reasoning based on your derivations.