Consider a spacecraft that is far away from planets or othermassive objects. The mass of the spacecraft is M = 1.5×105 kg. Therocket engines are shut off and the spacecraft coasts with avelocity vector v = (0, 20, 0) km/s. The space craft passes theposition x = (12, 15, 0) km at which time the spacecraft fires itsthruster rockets giving it a net force of F = (6 × 104 , 0, 0) Nwhich is exerted for 3.4 s. The ejected gases have total mass thatis small compared to the mass of the spacecraft. a) Where is thespace craft 1 hour afterwards? b) What approximations have you madein your analysis?

Kepler’s second law is this statement: A line segment joining aplanet and the Sun sweeps out equal areas during equal intervals oftime. We are going to prove this statement. Consider the wedge inthe figure with area dA = 1 2 R 2 d? The rate that area is sweptper unit time is dA dt = 1 2 R 2 d? dt = 1 2 R 2 ?? and this istrue even if radius R is varying. We take the origin to be thecenter of the Sun and radius R is the distance between planet andSun. The angle ? gives the position of the planet in the eclipticplane. Kepler’s second law is equivalent to dA dt = constant or d2A dt = 0. In class we showed that acceleration in polarcoordinates can be written a = (R¨ ? R ?? 2 )ˆr + (2R? ?? + R¨?)?ˆBecause the gravitational force is in the radial direction, thetangential component of acceleration is zero. This means that 2R??? + R¨? = 0 Show that this relation is equivalent to dA/dt =constant and Kepler’s second law.

A spherical hollow is made in a sphere of radius R = 11.3 cmsuch that its surface touches the outside surface of the sphere andpasses through its center (see Figure). The mass of the spherebefore hollowing was M = 57.0 kg. What is the magnitude of thegravitational force between the hollowed-out lead sphere and asmall sphere of mass m = 4.2 kg, located a distance d = 0.55 m fromthe center of the lead sphere?