Lesson 1 · from § of the 1910 text
Clock and Planet Problems
Problems for review
The book closes with a set of miscellaneous problems. They use no new principles: each is solved by the methods of the earlier chapters, chiefly the equation in one unknown, simultaneous equations, and the quadratic equation. The art lies in choosing the unknown and translating the words into an equation.
Clock problems
The face of a clock is divided into 60 minute spaces. The minute hand moves twelve times as fast as the hour hand: while the minute hand goes over 12 spaces, the hour hand goes over 1.
Example. At what time between 4 and 5 o'clock will the hands of a clock be together?
First, the minute hand moves twelve times as fast as the hour hand. Second, at 4 o'clock the hour hand is 20 minute spaces ahead of the minute hand. Now let x = the number of minute spaces that the minute hand travels from its position at 4 o'clock until it overtakes the hour hand. Obviously the hour hand must travel x - 20 spaces before it is overtaken by the minute hand.
| Therefore | x = 12(x - 20). |
| Whence | x = 21\frac{9}{11}. |
Hence the hands are together at 21\frac{9}{11} minutes after 4 o'clock.
The hands are in a straight line when they are together or when they point in opposite directions (30 spaces apart), and at right angles when they are 15 spaces apart. In each case find how many spaces the minute hand must gain on the hour hand.
Planets in opposition and conjunction
If the earth is between a planet and the sun and in a straight (nearly) line with them, the planet is said to be in opposition. If a planet is between the earth and the sun and in a straight (nearly) line with them, the planet is said to be in conjunction.
Example. The earth and Mars revolve about the sun in (approximately) 365 days and 687 days respectively. Mars was in opposition September 24, 1909. What is the approximate date of the next opposition?
For the sake of simplicity we will suppose in this and in similar problems that the planets move in the same plane and in circular paths of which the sun is the center. Let x = the required number of days. Now in x days the earth will make \dfrac{x}{365} revolutions about the sun, and Mars will make \dfrac{x}{687} revolutions. But to be in opposition the earth must in x days go round the sun once more than Mars does.
| Therefore | \dfrac{x}{365} = \dfrac{x}{687} + 1. |
| Whence | x = 778.7, nearly. |
Therefore the required date is 778 days later, November 11, 1911.
Give the number of days to one decimal place, or to the nearest day; the answer shows the date as well. (For a planet nearer the sun than the earth, such as Venus, the planet goes round once more than the earth.)
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Miscellaneous problems:
At what time between 4 and 5 o'clock will the hands of a clock be together?
At what time between 7 and 8 o'clock are the hands of a clock together?
At what times between 2 and 3 o'clock are the hands of a clock in a straight line?
At what time between 6 and 7 o'clock is the minute hand 10 minute spaces ahead of the hour hand? 10 minute spaces behind the hour hand?
At what times between 5 and 6 o'clock are the hands of a clock at right angles?
The earth and Mars revolve about the sun in (approximately) 365 days and 687 days respectively. Mars was in opposition September 24, 1909. How many days is it to the next opposition, and so what is its approximate date?
Venus was in (superior) conjunction April 28, 1909. If Venus revolves about the sun once in 225 days, find the number of days to the next conjunction, and so its approximate date.
Jupiter revolves about the sun once in 4332 days. Jupiter was in opposition February 28, 1909. Find the number of days to the next opposition, and so its approximate date.
Saturn revolves about the sun once in 10{,}759 days. It was in opposition April 3, 1909. Find the number of days to the next opposition, and so its approximate date.