A team of astronauts is on a mission to land on and explore a large asteroid. In addition to collecting samples and performing experiments, one of their tasks is to demonstrate the concept of the escape speed by throwing rocks straight up at various initial speeds. With what minimum initial speed Uesc will the rocks need to be thrown in order for them never to "fall" back to the asteroid? Assume that the asteroid is approximately spherical, with an average density p = 2.67 x 106 g/m³ and volume V = 1.71 × 10¹2 m³. Recall that the universal gravitational constant is G = 6.67 × 10-¹¹ N.m²/kg². Vesc = m/s

You are out on a date, eating dinner in a restaurant that has several television screens. Most of the screens are showing sports events, but one near you and your date is showing a discussion of an upcoming voyage to Mars. (a) Your date says, “I wonder how long it takes to get to Mars?” Wanting to impress your date, you grab a napkin and draw as shownon it. Even more impressively, you tell your date that the minimum-energy transfer orbit from Earth to Mars is an elliptical trajectory with the departure planet corresponding to the perihelion of the ellipse and the arrival planet at the aphelion. You pull out your smartphone, activate the calculator feature, and perform a calculation on another napkin to answer the question above that your date asked about the transfer time interval to Mars on this particular trajectory. (b) What If? Your date is impressed, but then asks you to determine the transit time to an inner planet, like Venus.

9 pl

Find the escape velocity ve for an object of mass m that is initially at a distance R from the center of a planet of mass M. Assume that R≥Rplanet, the radius of the planet, and ignore air resistance. Express the escape velocity in terms of R, M, m, and G, the universal gravitational constant.