Question

Launch a book from a cannon which is inclined at 45 degree to the horizontal and facing east. Write pertinent formulas and calculations that explains the motion of the book through space. Consider the effects due to: gravity near the surface of the Earth, air friction, the rotation of Earth and pseudo forces, the shape of the book and its rotation, universal gravitation, special relativity, and chaos theory. Consider a large range of initial velocities: from so small that the book just barely leaves the cannon to so large to consider special relativity.

Answer 1

all you need to know isKepler's first and third laws and the fact that the period of earth's (approximately circular) orbit is one year.

I am not sure I would say that the earth orbits the sun is because it has angular momentum. True, it has angular momentum, but the reason is that the sun exerts a force on it. Technically the straight line path is an orbit, but it has no angular momentum.

• Kepler's first law states that the orbit of a planet is an ellipse with a semimajor axis a and with the sun at one focus of the ellipse. A circular orbit has a semimajor axis equal to the radius of the circle, so a₁=R_E where R_E is the radius of the earth's orbit; the eccentricity of a circle is 0. The other extreme is an ellipse with eccentricity 1 which is a straight line from the sun to the earth and so the semimajor axis for a "dropped earth" is a₂=R_E/2. Of course, this is not an orbit you will actually see in nature because there is no such thing as a point mass and the speed of the earth when the two point masses coincide would be infinite, but if we can cleverly deduce the period of this orbit, one-half that period will be the answer to your question.

• Kepler's third law states that the square of the period T of an orbit is proportional to the cube of its semimajor axis, so T₁²/T₂²=a₁³/a₂³=R_E³/(R_E/2)³=8. So, T₂=T₁/√8=0.354 years. So, the time to go half a period is 0.177 years=64.6 days. I will let you put that into seconds.