## Answers to Example Questions for the Second Exam

1. c. As seen from mars, the planet is close to opposition (high in the sky at midnight). The planet has to be on an orbit outside Mars's. It could only be Saturn.

2. True. For and inferior planet, greatest elongation occurs when the angle Sun-Planet-Earth is 90 degrees. Sketch the configuration and see that sin (elongation) = (opposite)/(hypotenuse) = (Sun-Planet)/(Sun-Earth). Since sin (14.5) = 0.25, we see the statement is true. Note, when using your calculator, make sure the units of angle are degrees (and not radians).

3. d. Remember, retrograde motion occurs when the earth is 'overtaking on the inside track', ie when Mars is on the opposite side from the sun.

4. g. Synodic period is between consecutive identical configurations (eg opposition to opposition). So g is correct. Note that e is the definition for the Siderial period. Note that a is wrong because there are two quadratures (east and west) so from one to the next quadrature is NOT the full interval between identical configurations.

5. (1) Sunspots. The sun is not perfect, as required by Aristotle. (2) Mountains and craters on the moon. It seems too much like the earth to fit the Aristotelian picture of a celestial body. (3) The complete cycle of phases of venus confirm that it goes around the sun, not the earth. (4) Jupiter has 4 moons which obey Kepler's laws, so there are other things which are the center of motion, not just the earth.

6. He failed to see any parallactic shifts in the position of the new star (Nova) due to the nightly rotation of the earth, suggesting that it was at least further than the moon. This meant the distance to the Nova was greater than to the moon, placing it in the "superlunary" realm which was supposed to be perfect and unchanging according to Aristotle --- not the place to find a changing star.

7. look at your class notes. Draw the ellipse and label the perihelion, aphelion, sun at one focus, major diameter (the longest axis), and minor diameter (perpendicular smaller axis). To estimate the eccentricity, e=c/a where a is the semi-major axis, and c is the distance from the center of the ellipse to one focus.

8. Kepler's second law : a line from the sun to a planet sweeps out equal areas in equal times. (a) perihelion, (b) aphelion. This is because at perihelion the radial line from planet to sun is shortest, so the speed around the orbit must be fastest to generate the required area. Conversely, at aphelion, the radial line is longest, so to sweep out the same area, the speed around the orbit must be smallest.

9. c. First lets get the difference in radial distances between aphelion and perihelion. Recall that eccentricity is c/a (c is distance from center of ellipse to one focus, ie the sun; while a is the semimajor axis). So if e=c/a=0.5, then the sun is halfway between the center of the ellipse and the perihelion point. Hence perihelion distance Rp = a/2 while the aphelion distance Ra = a/2 + a = 3a/2 = 3Rp, so aphelion is three times further than perihelion. If we want the radial lines at perihelion and aphelion to sweep out equal areas in equal times, then the speed of the planet at perihelion must be three times greater than the speed at aphelion, since the radial line is only 1/3 as long at perihelion. So we have Sp = 3Sa = 30 km/s since Sa (speed at aphelion) is 10 km/s as stated in the question.

10. c. Remember, for planets orbiting the sun, P2=a3 as long as P is in years and a is in AU (Kepler's 3rd law). So a3=64. So a=4 (to take the cube root, remember it is the same as taking the 1/3 power).

11. c. Remember, speed is a scalar, velocity is a vector.

12. a. Use Newton's second law F=ma. For the same force, A gets twice the acceleration since it has half the mass of B.

13. d. Use Newton's law of gravity, which states that the gravitational force is proportional to the product of the masses and inversely proportional to the separation squared. Thus if the separation is halved the attraction becomes quadrupled.

14. e. With no sun, there is no force acting on the earth, so from Newton's 1st law the earth continues to move at constant velocity.

15. True. Weight is the gravitational force you exert on the floor or the weigh-scales. You can change this "easily" by going to another planet, but you would look just the same. Your ultimate aim is to look "thinner", and this can only be done by losing some mass. (Recall "mass" is an intrinsic property of you, independent of where you are, and is a measure of the amount of "stuff" in you.

16. First, the formula for acceleration due to gravity, gE, at the surface of the Earth, is gE = GM/R2, where M and R are the mass and radius of the earth. Hence, for planet A which has double M but the same R, the acceleration due to gravity on A is simply TWICE that on earth, or 20 m/s/s. More explicitly : gA = Gx2M/R2 = 2 x gE = 20 m/s/s. Next is planet B which has the same mass as A but double its radius. Hence, when you stand on the surface of B you are TWICE as far from the center and hence the gravitational acceleration is A QUARTER. So gB = 1/4 x gA = 1/4 x 2 x gE = 1/2 x gE = 5 m/s/s. More explicitly, compared to the earth, planet B has double the mass and double the radius, so gB = Gx2M/(2R)2 = 2/4 x GM/R2 = 1/2 gE = 5 m/s/s.

17. False. The space shuttle is in orbit, ie it is in 'free fall' --- it needs no rocket engines to stay up.

18. True. Hyperbolic orbits are open and not closed like an ellipse, we say the object is "unbound". Examples of such orbits include passing stars.