Practice Graviation Problems
1. Consider the 1200 kg rocket changing from one orbit with a radius of R1 to an orbit of radius R2, as shown in the figure below. More specifically, you are trying to place a rocket which is originally traveling around the Sun alongside the Earth (orbit radius R1 traveling at the same velocity as the Earth) into a corresponding orbit with the same radius as Mars (R2).
a) Find the speed of the rocket when it is at the position of the Earth originally. (Remember v=d/t and the rocket has the same period as the Earth when it is at R1).
b) Ignoring the mass of the planets, find the force on the satellite at R1.
c) What is the mechanical energy of the rocket at R1?
d) If there is no energy added by booster rockets, then calculate the speed of the rocket when it gets to the orbit of Mars.
e) What is the force on the satellite when it reaches Mars' orbit (again ignoring the influence of the planet masses)?
f) What speed is necessary in order for the rocket to achieve a stable circular orbit of radius R2?
2. How much energy is expended in putting a 10,000 kg space ship into a stable circular orbit whose period is 72 hours around the earth?
(Hint: You can find the radius of the orbit by using Kepler's 3rd law
T2 = R3 . The mass of the earth is Me = 6x1024 kg, the radius of the earth is Re = 6.4 x106 and G=6.67x10-11 N m2/kg2.)
3. Consider an asteroid of mass 2x1024 kg traveling toward a star as shown below.
a) What is the force exerted on the asteroid when the asteroid gets to within R1=1x109 m.
b) By defining Ugrav=0 at infinite separation, calculate the total mechanical energy of the asteroid for the position described above.
c) Find the final asteroid velocity if it initially starts at R1 from rest and ends up at the surface of the star.
d) The asteroid embeds itself in the star. If all of the initial mechanical energy of the asteroid is changed into rotational energy, calculate the angular velocity of the star.
4. Consider the system of the Sun, Earth, and Venus, as shown below. As an approximation, assume each planet revolves around the Sun in a circular orbit. The distance between the Sun and Earth is 1.5x1011 m and the distance from the Sun to Venus is 1.0x1011 m. The masses of the two planets are 6x1024 kg and 4.9x1024 kg for the Earth and Venus, respectively. The two planets are lined up in opposition as shown in the diagram.
a) Calculate the force on the Earth due to the sun only (MS = 2x1030).
b) Again ignoring Venus, find the centripetal acceleration and the speed of the Earth in its orbit.
c) Now consider the mass of Venus. Assuming the same velocity for the Earth found above, by how much will the radius of the Earth's orbit change because of the extra gravitational pull of Venus if it is positioned as shown above? (Note that this causes a variable perturbation in the orbit of the Earth as Venus revolves around the Sun.)
5. Your space ship is caught by the gravity of the sun and you must decide how long to speed up in order to escape.
a) By using the description of the gravitational potential where U(r)=0 infinitely far from the sun, calculate the potential energy of a 1000 kg rocket which is 5x106 m from the center of the sun.
b) What is the final mechanical energy for the rocket if it just escapes the gravitational field of the sun with zero velocity?
c) Calculate the minimum speed necessary to escape the sun's gravitational attraction.
d) What is the force of attraction between the rocket and the sun for the 1000 kg space ship located 5x106 m from the center of the sun?
6. Mars Lander. So it is your job to land a space probe on the surface of the planet Mars. The masses and distances are given below. Assume that Mars rotates with a period of 10 hours.
a) Calculate the speed of a point on the surface of Mars located at the equator.
b) What is the potential energy of the satellite due to Mars when it is at the position shown above?
c) Calculate the force on the satellite when it is in the position shown above.
d) You now wish to land the satellite on the planet starting from the position shown. In order to do so, you choose an elliptical path starting with some initial (tangential) velocity Vo. Find the initial velocity Vo required to land the probe on the equator so that it has the same speed as the planet’s surface (the answer found to part a)).