Question

Consider a spacecraft that is far away from planets or other massive objects. The mass of the spacecraft is M = 1.5×10^5 kg. The rocket engines are shut off and the spacecraft coasts with a velocity vector v = (0, 20, 0) km/s. The spacecraft passes the position x = (12, 15, 0) km at which time the spacecraft fires its thruster rockets giving it a net force of F = (6 × 10^4, 0, 0) N which is exerted for 3.4 s. The ejected gases have a total mass that is small compared to the mass of the spacecraft. a) Where is the spacecraft 1 hour afterwards? b) What approximations have you made in your analysis? Kepler's second law is this statement: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. We are going to prove this statement. Consider the wedge in the figure with area dA = 1/2 R^2 dθ. The rate that area is swept per unit time is dA/dt = 1/2 R^2 dθ/dt = 1/2 R^2 θ˙ and this is true even if radius R is varying. We take the origin to be the center of the Sun and radius R is the distance between the planet and the Sun. The angle θ gives the position of the planet in the ecliptic plane. Kepler's second law is equivalent to dA/dt = constant or d^2A/dt = 0. In class, we showed that acceleration in polar coordinates can be written as a = (R¨ - Rθ˙^2)r̂ + (2Ṙθ̇ + Rθ̈)θ̂. Because the gravitational force is in the radial direction, the tangential component of acceleration is zero. This means that 2Ṙθ̇ + Rθ̈ = 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 cm such that its surface touches the outside surface of the sphere and passes through its center (see Figure). The mass of the sphere before hollowing was M = 57.0 kg. What is the magnitude of the gravitational force between the hollowed-out lead sphere and a small sphere of mass m = 4.2 kg, located a distance d = 0.55 m from the center of the lead sphere?

          Consider a spacecraft that is far away from planets or other massive objects. The mass of the spacecraft is M = 1.5×10^5 kg. The rocket engines are shut off and the spacecraft coasts with a velocity vector v = (0, 20, 0) km/s. The spacecraft passes the position x = (12, 15, 0) km at which time the spacecraft fires its thruster rockets giving it a net force of F = (6 × 10^4, 0, 0) N which is exerted for 3.4 s. The ejected gases have a total mass that is small compared to the mass of the spacecraft. a) Where is the spacecraft 1 hour afterwards? b) What approximations have you made in your analysis?

Kepler's second law is this statement: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. We are going to prove this statement. Consider the wedge in the figure with area dA = 1/2 R^2 dθ. The rate that area is swept per unit time is dA/dt = 1/2 R^2 dθ/dt = 1/2 R^2 θ˙ and this is true even if radius R is varying. We take the origin to be the center of the Sun and radius R is the distance between the planet and the Sun. The angle θ gives the position of the planet in the ecliptic plane. Kepler's second law is equivalent to dA/dt = constant or d^2A/dt = 0. In class, we showed that acceleration in polar coordinates can be written as a = (R¨ - Rθ˙^2)r̂ + (2Ṙθ̇ + Rθ̈)θ̂. Because the gravitational force is in the radial direction, the tangential component of acceleration is zero. This means that 2Ṙθ̇ + Rθ̈ = 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 cm such that its surface touches the outside surface of the sphere and passes through its center (see Figure). The mass of the sphere before hollowing was M = 57.0 kg. What is the magnitude of the gravitational force between the hollowed-out lead sphere and a small sphere of mass m = 4.2 kg, located a distance d = 0.55 m from the center of the lead sphere?

Added by Ryan S.

Question

Please give Ace some feedback