**Concepts:**

- units conversion
- geometry
- division and multiplication
- graphing
- analyzing data
- using equations

When we think of how fast we're going (or the ** rate** at which we're traveling)
in a car, we usually express it in "miles per hour". This ** rate** (miles/hr) is
really reflecting the average distance that we've gone in the time it takes to travel
that distance. For example, a car that travels 60 miles in 1 hour is traveling at a rate
of 60 miles/hr. Alternately, a car that has a speed of 60 miles/hr will have traveled
60 miles in 1 hour. At this rate, in 2 hours it will have traveled a distance of 120 miles.

Therefore, given
some basic information, like how fast an object is going (its speed
or **rate**)
and how long it took to get to where it is now (** time**),
it is possible to determine the distance that object has traveled.
This concept can be expressed as an equation that has the following form:

where ** D ** is the distance, ** R ** is the rate or speed, and ** T **
is the elapsed time.

This basic idea can be applied to many situations. In the following exercises, you
will calculate the distance to the moon (** TASK A**), and the
speed or ** rate ** of the Galileo Spacecraft on its way to
Jupiter (**TASK B **).

For the calculations we will perform in this lesson, we will be
using the ** metric ** system. That is, we'll express **
distance ** in terms of
** kilometers ** and * not * ** miles **.

** Some useful relationships: **

- 1 mile = 5280 feet
- 1 mile = 1.61 kilometer
- 1 kilometer = 0.62 miles
- 1 hour = 60 minutes
- 1 minute = 60 seconds
- 1 hour = 60 minutes/hour x 60 seconds/minute = 3600 seconds
- 24 hours = 1 day

The Apollo 11 spacecraft was launched from Cape Kennedy at 13:31:01
GMT on July 16, 1969. After ** 2 hr and 33 min ** in Earth orbit,
the S-IVB
rocket engine was
reignited for acceleration of the spacecraft to the velocity required to
escape Earth's gravity. Although at times the spacecraft reached speeds
near 40,000 km/hr, the average speed was about ** 5500 km/hr **.

Lunar-orbit insertion began at 75:50 ** (75 hours and 50 minutes)
**
ground elapsed time (GET). The spacecraft was placed in an elliptical
orbit (61 by 169 nautical miles), inclined 1.25
degrees to the lunar equatorial plane. At 80:12 GET, the service module
propulsion system was reignited, and the orbit was made nearly circular
(66 by 54 nautical miles) above the surface of the Moon. Each orbit took
two hours.

The lunar module (LM), with Astronauts Armstrong and Aldrin aboard, was undocked from the command-service module (CSM) at 100:14 GET, following a thorough check of all the LM systems. At 101:36 GET, the LM descent engine was fired for approximately 29 seconds, and the descent to the lunar surface began. At 102:33 GET, the LM descent engine was started for the last time and burned until touchdown on the lunar surface. Eagle landed on the Moon 102 hr, 45 min and 40 sec after launch.

**ACTIVITIES:** Using the above information and our ** D = R x T **
equation, can you calculated the distance to the moon?

- The
**Elapsed Time**= (75 hrs 50 min)- (2 hrs 33 min) =**73 hrs 27 min** - average speed
**(Rate)**=**5500 km/hr**

The

If you would like to learn more about the ** Apollo 11 ** mission to
the moon, try
Apollo Facts .

Path of Spacecraft

If you'd like to learn more about navigating Galileo to Jupiter, check out:
JPL-Trajectory

Date
| Elapsed Time(hours) | Distance from the Earth(millions of km) |
Distance Traveled(millions of km) | Rate(km/sec) | ||

Starting Point | 1-1-95 | 0 | 883.7 | --- | --- | |
---|---|---|---|---|---|---|

Position 1 | 1-5-95 | 96 | 880.4 | 3.3 | 9.6 | |

Position 2 | 1-13-95 | 288 | 872.2 | 11.5 | 11.1 | |

Position 3 | 1-18-95 | 408 | 866.1 | 17.7 | --- | |

Position 4 | 1-25-95 | 576 | 856.3 | 27.4 | 13.2 | |

Position 5 | 1-31-95 | 720 | 846.8 | 36.9 | --- |

**ACTIVITIES:** Using the above table, which tracks the distance
of the Galileo
Spacecraft from the earth in January of last year, make two graphs:

- The
**Elapsed Time**(Convert hours to days.)*vs.***Distance**from the Earth, and - the
**Date***vs.*the**Distance Traveled.**

** Sample Plot: **

** Interpretation: **

- At this time in its mission (January 1995), the Galileo Spacecraft is circling (in orbit) around Jupiter.
**Q:**Looking at your first graph, is Galileo traveling*away from*or*towards the Earth*?**A:**It is traveling*towards the earth*, since the**Distance from the Earth**is getting smaller as time progresses.**Q:**What is the total distance Galileo has traveled in the month of January?**A:**36900000 km (or 36.9*million km's*)**Q:**What trends do you notice in looking at your graph of**Date***vs.*the**Distance Traveled?****A:***The distance traveled is not constant, indicating that the spacecraft is speeding up.***Q:**Calculate the**Rate**for**Positions 3**and**5**in the above table.**A:***12 km/sec*and*14.2 km/sec***HINT:**Since**rate**is**distance**per**time**, divide the**Distance traveled**by the**Elapsed Time**. Remember to convert*hours*to**seconds**, and that the values in the table are*millions*of km's.**Sample Calculation,**Position 2 :

How was the**rate**for*Position 2*in the table calculated? Since distance divided by time is rate, we use the**Distance Traveled**and divide by**Elapsed Time**:

**Rate**= (11500000 km) / [(288 hours) x (60 min/hr) x (60sec/min)] = 11.1 km/sec

This module was written by

NOAO/ Planetary Sciences, University of Arizona, Tucson AZ.

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Galileo Solid State Imaging Team Leader: Dr. Michael J. S. Belton

The SSI Education and Public Outreach webpages were originally created and managed by Matthew Fishburn and Elizabeth Alvarez with significant assistance from Kelly Bender, Ross Beyer, Detrick Branston, Stephanie Lyons, Eileen Ryan, and Nalin Samarasinha.

Last updated: **September 17, 1999**, by Matthew Fishburn

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