What is Speed?

DEFINITION 5. Speed

21. Every body that is moving has a speed or velocity.

This is measured by the distance that body traverses in equal intervals of time, with a uniform motion.

When body B travels twice the distance at a uniform speed that body A travels through in the same time, also moving uniformly, then body B is said to have a speed twice as great as body A.

Corollary 1.

22. A body in uniform motion will traverse equal distances in equal intervals of time (17).

And so, the body that has moved uniformly will have the same speed or velocity always.

Corollary 2.

23. a. The speed that a body has at some point in the distance traversed with a non-uniform motion, is to be measured by the distance that the body would traverse in a given time with that uniform speed.

Corollary 3.

b. The speed of a body with absolute uniform motion can be measured from the distance that the body traverses, for example, in a single second.

The speed of the same perfectly well-known body is agreed upon, which prevails to be defined as the distance which that body travels through in a time of one second.

Scholium.

24. This ratio is also especially useful in the measurement of speed.

Sailors find the speed of the ship by measuring the distance that the ship travels in a given time.

Commonly they find how many miles a ship has traveled in 1/4 of an hour if it were in a uniform motion.

PROPOSITION 2. Theorem.

25. For two bodies progressing with uniform motion, the speeds vary directly with the distances traveled, and inversely with the time with which these distances are traversed.

Demonstration

Body A travels at speed C.

• It travels through a distance S in time T

Body a travels at speed c.

• It travels at distance s in time t.

They are in uniform motion. The distances are in proportion to the times (18), the distance sT

that body a completes in time T can be determined from the proportion t : T = s : t : sT

Therefore in a time T, the body a will move through a distance t.

But body A in the same time T moves through a distance S. The speeds of the bodies ought to be measured

from the distances the bodies travel in the same time (18). On account of which C : c = S

sT

t , or C : c = TS : st . From which it follows that the speeds are directly as the distances and inversely as the times, with which these are traversed. Q. E. D.

Corollary 1.

CT ct

26. The equation S = s is produced from the final ratio. In any uniform motion therefore the product of the speed and the time, if divided by the distance traveled in that time, always gives the same quotient.

Corollary 2

27. Also there is the ratio T : t = CS : cs .

It follows that the times are in the ratio of the direct proportions of the distances and in the inverse ratio of the speeds, or to be as the distance divided by the speed.

Corollary 3

28. Then the proportion found can also be changed into this S : s = CT : ct. From which the distances traversed with the uniform motion are gathered together to be in a ratio composed from the ratio of the speeds and from the ratio of the times.

Corollary 4

29. Therefore with the speed of the body given, moving uniformly, together with some distance described, the time can be noted in which this distance has been traversed; clearly by dividing the distance by the speed. Since indeed we have shown that this amount is always in proportion to the time, we can take the same as a measure of time.

Corollary 5.

30. Similarly the speed can be expressed by the distance traveled through divided by the time, and the distance also by the product of the time and the speed.

Scholium 1.

31. For if the speed is such that, in order that the moving body completes three feet in one second, and therefore we can call the speed by the number 3; we will try to find the time in which for example 60 feet in the same motion are completed. For 60 divided by 3 and the quotient is 20 will show that it takes 20 seconds for the motion to be traversed.

If the distance is sought for the time of 12 seconds to be traversed, the product will be 36 feet. And also for a body traveling 4.8 feet in 6 seconds, the speed that arises is 8, which shows that this body travels 8 in one second.

Scholium 2.

32. Hence, the times, the distances, and the speeds are to be measured in the following ratios that we will always adhere to.

We will always express:

• time in seconds
• distance in Rhenish feet
• speed is in the number of feet traversed per second

It is more convenient for the speed to be determined from the units that it will encounter, which henceforth we will be using, but these units arises from those, and the former can easily be recalled.