**Questions 3 and 4 below refer to the universal gravitational potential energy function \( U(r) \) for a planet with mass \( M \) and radius \( R \), which is shown in the figure below.****3.**If a projectile is launched from the surface of the planet with a speed equal to the planet's escape speed, how much mechanical energy does it have when it reaches a distance \( r = \infty \)?**Graph Explanation:**- The graph depicts the gravitational potential energy \( U(r) \) as a function of distance \( r \) from the center of the planet.- On the y-axis, \( U(r) \) decreases steeply from a value at a distance \( r = R \) from the planet's center and approaches zero as \( r \to \infty \).- The curve is hyperbolic, indicating a negative potential energy field.**4.**If an astronaut who is orbiting the planet at a radius of \( 3R \) wants to move their orbit to a radius of \( 2R \), what direction should they fire their thrusters? (and how is your answer supported by the given graph of \( U(r) \)?)

Since you have asked multiple question, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question. 3. The potential energy of an object due to the gravitational pull of a planet is given by the function: Ur=-GMmrwhere G is the universal gravitation constant, M is the mass of the planet, m is the mass of the object and r is the distance of the object from the planet. The kinetic energy of an object is given by the formula: K=12mv2 When the object moves at a speed equal to the planet's escape velocity, the velocity is given by: vesc=2GMr Hence, the kinetic energy of the object moving at the escape velocity is given as: K=12mv2=12m2GMr2=12m2GMr=GMmrThe total mechanical energy is given by: T=K+U Substitute the relations for U and K obtained in step 1 to calculate the total mechanical energy of the object: T=K+U=GMmr+-GMmr=GMmr-GMmr=0 The total mechanical energy is independent of r and is constant. The total mechanical energy is 0.