Lesson 2 · from §41 of the 1910 text
Uniform Motion
A train traveling 8 hours at 40 miles an hour goes 8 \times 40 = 320 miles. That's uniform motion, tying together time t, rate r, and distance d: d = r \times t. A huge number of problems in algebra and physics rest on this one equation.
Example. A walker going 4 miles per hour is overtaken 14 hours after leaving a point by a horseman who left the same point 8 hours after the walker. Find the horseman's rate.
Each one travels a different time at a different rate, but both cover the same distance. A table keeps it straight:
| time t (hr) | rate r (mi/hr) | distance d = rt | |
|---|---|---|---|
| Walker | 14 | 4 | 56 |
| Horseman | 6 | x | 6x |
So 6x = 56 and x = 9\tfrac13 miles per hour. Check: 4 \cdot 14 = 56 and 9\tfrac13 \cdot 6 = 56. ✓
Quick check
Two cyclists leave the same spot in opposite directions at 9 and 11 miles per hour. In how many hours are they 100 miles apart?
From the history of algebra
Sir Isaac Newton (1642-1727) was probably the keenest mathematical thinker who ever lived. He was the son of a farmer of slender means, and as a boy he was rather lazy. The story goes that after he beat a bigger boy in a schoolyard fight, it struck him that he might do just as well in his studies if he really tried, and he never let up after that. In his Principia he showed that everything, great and small, from a planet circling the sun to an apple falling from a tree, follows the same laws of motion. He also invented the calculus. His biggest gift, though, may have been an idea: that every particle of matter is connected with every other, and we simply occupy our place in a universe run by universal law. (A copy of the Principia that Newton himself presented to Yale is still in Yale's library.)
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.
Uniform motion problems. Make a table of time, rate, and distance.
A and B start from the same place at the same time and travel in opposite directions. A goes 8 miles per hour and B 10. In how many hours will they be 120 miles apart? 180 miles apart?
(Same setup: same place, opposite directions.) A travels twice as fast as B. In 5 hours they are 135 miles apart. Find each rate.
(Opposite directions.) A travels 2 miles per hour more than B. After 8 hours they are 96 miles apart. Find each rate.
(Opposite directions.) A goes 4 miles per hour more than B. After 6 hours they are 168 miles apart. Find each rate.
(Opposite directions.) B goes 3 miles per hour less than A, and travels \tfrac34 as fast as A. Find each rate. After how many hours will they be 168 miles apart?
(Opposite directions.) A travels 3 hours and stops; B travels 5 hours. Then they are 77 miles apart. A's rate is twice B's. Find their rates and how far each traveled.
(Opposite directions.) B travels 9 hours at 4 miles per hour less than A. A covers the same distance in 3 hours less time, then stops. How far apart are they after 9 hours?
A and B start at the same time from two points 144 miles apart and travel toward each other until they meet. They travel at the same rate and meet in 8 hours. Find each rate.
(144 miles apart, toward each other.) A travels 2 miles per hour less than B, and they meet in 9 hours. Find each rate.
(144 miles apart, toward each other.) B travels three times as fast as A, and they meet in 12 hours. Find each rate.
(144 miles apart, toward each other.) They meet in 6 hours, and B travels 24 miles more than A. Find each rate.
(144 miles apart, toward each other.) They meet in 8 hours, and B goes 2 miles per hour more than A. Find each rate.
(144 miles apart, toward each other.) They meet in 9 hours, and A travels 4 miles per hour more than B. Find each rate.
(144 miles apart, toward each other.) B goes 6 miles per hour more than A and travels twice as far. How many hours until they meet?
(144 miles apart, toward each other.) A travels 6 miles per hour and B 9, but B is delayed 4 hours on the way. How many hours from the start until they meet?
(144 miles apart, toward each other.) A is delayed 3 hours and B 5 hours; their rates are 7 and 9 miles per hour. How many hours from the start until they meet?
A and B start from the same place at the same time and travel in opposite directions, A at 3 and B at 4 miles per hour. In how many hours will they be 42 miles apart?
A and B start at the same time from two points 72 miles apart and travel toward each other, A at 8 and B at 10 miles per hour. In how many hours will they meet?
Kansas City is 285 miles from St. Louis. A train running 37 miles per hour leaves Kansas City for St. Louis just as a train running 38 miles per hour leaves St. Louis for Kansas City. In how many hours will they meet?
A starts from a place and travels 8 miles per hour. Four hours later B starts from the same place in the same direction at 10 miles per hour. How many hours does B travel before overtaking A?
Two bicyclists 108 miles apart start at the same time toward each other. One travels 10 miles per hour, the other 12, but the second is delayed 2 hours on the way. In how many hours will they meet, and how far has each traveled?
A passenger train starts 2 hours after a freight train, from the same station but in the opposite direction. The passenger train goes 42 miles per hour and the freight 24. How many hours after the passenger train starts will they be 246 miles apart?
A messenger going 8 miles per hour has traveled 2 hours when it's found necessary to change the message. How fast must a second messenger travel to overtake the first in 6 hours?
A passenger train and a freight train start together from the same station in the same direction on parallel tracks at 45 and 18 miles per hour. How many hours until the passenger train is 144 miles ahead?
P is 108 miles from Q. A and B leave P together toward Q; whoever reaches Q first turns right back. A's rate is 9 miles per hour and B's is 15. How far has each traveled when they meet?
(P to Q is 108 miles, the first to reach Q turns back.) B travels five times as fast as A. How far has each traveled when they meet?
(P to Q is 108 miles, the first to reach Q turns back.) A travels 56 miles more than B. How far has each traveled when they meet?
(P to Q is 108 miles, the first to reach Q turns back.) They meet 24 miles from Q, and B is faster. How far has each traveled?
(P to Q is 108 miles, the first to reach Q turns back; A is faster.) They meet in 6 hours halfway between P and Q. Find each rate.
(P to Q is 108 miles; A is faster.) They meet in 6 hours, \tfrac23 of the way from P to Q. Find each rate.
(P to Q is 108 miles; A is faster.) They meet in 6 hours, 12 miles from Q. Find each rate.
(P to Q is 108 miles; A is faster.) They meet in 6 hours, 96 miles from P. Find each rate.
(P to Q is 108 miles.) A travels 4 miles per hour more than B and meets B in 12 hours. Find each rate.
(P to Q is 108 miles.) They meet in 6 hours, and A travels 2 miles per hour more than B. Find the distance each travels.
(P to Q is 108 miles.) A travels 4 miles per hour more than B, and they meet in 9 hours. Find the distance each travels.
Take sound as 1100 feet per second. Two and a half seconds after a marksman fires, he hears the bullet strike a target 550 yards away. Find the bullet's average velocity in feet per second.
(Sound: 1100 ft/s.) One and three-fourths seconds after a marksman fires his revolver, he hears the bullet strike a target 50 rods away. Find the bullet's average velocity in feet per second. (A rod is 16\tfrac12 feet.)
(Sound: 1100 ft/s.) A marksman fires at a target 1000 yards away. The bullet, averaging 1650 feet per second, passes over a boy who hears the strike and the gun's report at the same instant. How far is the boy from the target, in feet?