Chapter 8, Problem 8.38P

Imagine that the entire Sun collapses to a sphere of radius R g such that the work required to remove a small mass m from the surface would be equal to its rest energy mc 2. This radius is called the gravitational radius for the Sun. Find R g . (It is believed that the ultimate fate of very massive stars is to collapse beyond their gravitational radii into black holes.)

A particle is moving on xy plane, with x=15sin(2t), and y = 20cos(2t) In what kind of orbit is this particle moving?

Show that the values vA and vP of the speed of an earth satellite at the apogee A and the perigee P of an elliptic orbit are defined by the relationswhere M is the mass of the earth, and rA and rP represent, respectively, the maximum and minimum distances of the orbit to the center of the earth.

in order for a subatomic particle from space with a lifetime of τ in its own reference frame to reach the Earth’s surface, it must be able to travel a distance greater than or equal to the thickness of the atmosphere before it decays. If the thickness of the atmosphere i d in the Earth’s reference frame, find the minimum speed v at which this particle must travel in order to reach the Earth’s surface.

On the surface of the earth, the gravitational field (with z as vertical coordinate measured in meters) is F =〈0, 0, −g〉.(a) Find a potential function for F.  (b) Beginning at rest, a ball of mass m = 2 kg moves under the influence of gravity (without friction) along a path from P = (3, 2, 400) to Q = (−21, 40, 50). Find the ball’s velocity when it reaches Q.

Calculate the gravitational constant g, in SI units, for alocation on the surface of the sun.

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 vesc 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 ? = 2.67 × 106 g/m3 and volume V =1.71 × 1012 m3. Recall that the universal gravitational constant is G = 6.67 × 10-11 (Nm2)/(kg2).

Find the escape speed of a projectile from the surface of Jupiter.

Repeat the preceding problem with the ship heading directly away from the Earth.