ALGEBRA

PROBLEM 5.

PROBLEM 5. a. How long did Pioneer lO's radio signals, traveling at the speed of light (3.00 x 10^5 km/s) take to reach Earth from the distance of Pluto in April 1983 (4.58 x 10^9 km)?

Solution:

time = distance/speed = (4.58 x 10^9/3.00 x 10^5) s
=1.53 x 10^4 s
= 1.53 x10^4/3600) h, or 4.25 hours

b. What was Pioneer lO's average speed, in km/h, if it traveled about 4.58 x 10^9 km between 3 March 1972 and 25 April 1983?

Solution: From 3 March 1972 to 3 March 1983 there were 11 years, of which 2 were leap years, and from 3 March 1983 to 25 April there were another 53 days. The time for Pioneer 10 to travel that distance was (365 x 11) + 2 + 53 = 4070 days, or 4070 x 24 = 97680 hours.

Average speed = (4.58 x 10^9/9.77 x 10^4) km/h = 4.69 x 10^4 km/h

(We note that the average speed over this period is less than the average speed over the 21-month period of Problem 6 in Chapter 2.)

The time required for an orbiting satellite to make one complete revolution of Earth is called its "period." The length of the period depends on the location of the observer making the measurement.

Suppose the observer is located far out in space and views the satellite against the background of fixed stars. The period measured in this manner is called the "sidereal" period of revolution, or the period in relation to the stars. Note that the rotation of Earth does not affect the sidereal period. Now suppose that the observer is standing on Earth's equator. A satellite is overhead in low Earth orbit moving directly eastward. When the satellite has made one complete transit of its orbit, it will not yet be overhead for the observer because the rotation of Earth will have carried the observer a distance eastward. The satellite must travel an additional distance to again be over the observer's head. The observer measures the period of the satellite as the time elapsing between successive passes directly overhead. This period is called the "synodic" period of revolution, or the period between successive conjunctions, and it takes into account the rotation of Earth.

Spacecraft usually orbit in the same easterly direction as Earth's rotation: this is called a "posigrade" orbit. All U.S. manned spaceflights have been launched in posigrade orbits to take advantage of the extra velocity given to the spacecraft by Earth's rotation. In this case, the synodic period is greater than the sidereal period.

If the direction of orbiting is westerly, or opposite to Earth's rotation, the orbit is said to be "retrograde." In this case, an Earth observer would meet the satellite before it made one complete revolution around Earth, and the synodic period would be less than the sidereal period.

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