Part A Consider a uniform gravitational field (a fair approximation near the surface of a planet). Find U(yf)−U(y0)=−∫yfy0F⃗ g⋅ds⃗ , where F⃗ g=−mgj^ and ds⃗ =dyj^. Express your answer in terms of m, g, y0, and yf. U(yf)−U(y0)

Part A Consider a uniform gravitational field (a fair approximation near the surface of a planet). Find U(yf)−U(y0)=−∫yfy0F⃗ g⋅ds⃗ , where F⃗ g=−mgj^ and ds⃗ =dyj^. Express your answer in terms of m, g, y0, and yf. U(yf)−U(y0) =
Part B Consider the force exerted by a spring that obeys Hooke’s law. Find U(xf)−U(x0)=−∫xfx0F⃗ s⋅ds⃗ , where F⃗ s=−kxi^,ds⃗ =dxi^, and the spring constant k is positive. Express your answer in terms of k, x0, and xf. U(xf)−U(x0) =
Part C Finally, consider the gravitational force generated by a spherically symmetrical massive object. The magnitude and direction of such a force are given by Newton’s law of gravity: F⃗ G=−Gm1m2r2r^, where ds⃗ =drr^; G, m1, and m2 are constants; and r>0. Find U(rf)−U(r0)=−∫rfr0F⃗ G⋅ds⃗ . Express your answer in terms of G, m1, m2, r0, and rf. U(rf)−U(r0) =
Suppose our experimenter repeats his experiment on a planet more massive than Earth, where the acceleration due to gravity is g=30 m/s2. When he releases the ball from chin height without giving it a push, how will the ball’s behavior differ from its behavior on Earth? Ignore friction and air resistance. (Select all that apply.)