concrete number (grams per cubic centimetre), and the latter an abstract number, as it expresses a ratio only. Density and specific gravity being numerically the same when the metric system is employed, a Table of Specific Gravities serves also as a Table of Densities, and the word density comes to be used in place of specific gravity. CHAPTER IV. MOTION, INERTIA, AND FORCE. 25. Motion and Velocity.-Motion is change of position in relation to surrounding objects. A point or a body is said to be in motion when it has different positions at different times. To understand the motion of a body, we must know the speed of the body and the direction or path in which it moves. At present we shall consider motion in a straight line, that is, linear, or, more correctly, rectilinear motion (Lat. rectus, straight ; linea, a line). Uniform rectilinear motion is motion along a straight line with uniform or constant speed. Velocity is a certain speed in a definite direction, i.e. velocity is rate of motion. It is measured when uniform by the amount of motion, expressed in units of length, which takes place in a unit of time. A body moving in a straight line is said to move with uniform linear velocity when it passes through equal distances in equal times, however small these times may be. It is important to notice the last words of the definition of uniform linear velocity. A train might move through a mile in one minute and through another mile in the next minute, and yet it might have passed over unequal distances in successive seconds. When a body does not pass over equal distances in equal times, however small these intervals may be, its velocity is said to be variable. To measure velocity, we require a standard or unit of velocity. The unit of velocity in the British system is the velocity of a body which passes over unit length in unit time, i.c. a velocity of one foot per second. In the metric or C.G.S. system, the unit of velocity is the velocity of a body that describes one centimetre in one second. In all cases of velocity the unit of distance and the unit of time must both be expressed, unless they are clearly understood. Other units may be employed, as miles per minute, or metres per second, and they can be readily reduced to the standard units if required. Example.-Express a velocity of 10 miles per hour in British units of velocity, i.e. in feet per second. 10 miles per hour may be written 10 miles This is equal to I hour If v represent the number of units of velocity described by a body moving with uniform velocity in a unit of time, then the distance or space described in two units of time will be v × 2, in three units of time v × 3, in units of time vxt, i.e. vt. Representing the space by s, we see that the space described in one second with the velocity v is v, in two seconds v × 2, in three seconds v × 3, and in vxt. Hence the algebraic formula— or the expression s = vt Space velocity X time = seconds This general equation or formula of calculation for all cases of motion with uniform velocity applies whether we measure in British or metric units, and it is true whatever unit we employ, provided we keep to the same units throughout the particular example. Example.-A body moves at the rate of 50 feet per second. How far does it move in 5 minutes? Using the formula s = vt, we have v = 50 feet per second, and t = (We must reduce to seconds, as the velocity is given From s = vt we get here 5 mins. = 300 secs. in feet per second.) Example.-A train travels at a speed of 20 kms. an hour. How far would it go at this speed in 25 mins.? t= Taking the kilometres and the hour as units, we have v = 20 kms. and hour. Hence from s vt we get From the equation s = vt we get, by dividing both sides This enables us to calculate the velocity when space and time are given. We can also obtain from the equation s = vt by This enables us to calculate time when space and velocity are given. Example. Find the time required to move through a distance of 1 mile at the uniform velocity of 88 feet per second. It will be useful to remember that a velocity of a mile a minute is equal to 88 feet per second. The above examples may also be worked by the rules of simple proportion. 26. A body moving in a straight line may have more than one velocity at the same time. A boat that is being rowed down a stream has a velocity made up of the velocity produced by the rowing and the velocity of the stream. Both velocities are in the same direction, and the velocity of the boat is the sum of the two velocities. Rowed up stream the two velocities are in the opposite direction, and the resultant velocity is the difference of the two velocities in the direction of the greater. Example.-A stone rolls along a plank due south at the rate of 15 yards a minute, and the plank is being carried due north at the rate of 12 yards a minute. What is the resultant velocity of the stone in foot second units? The velocities being in opposite directions, and the greater velocity being south, the resultant velocity is 15 - 12 = 3 yards a minute due south. 27. Variable Velocity.-When a body is moving with variable velocity, i.e. when a body moves over unequal spaces in equal times, we may not be able to give its velocity at any instant, yet we can find its average or mean velocity for any distance traversed, by dividing that distance by the whole time taken to traverse it. A train that runs 180 miles in six hours, though its actual velocity may vary from o to 60, runs the distance with an average velocity of 180 = 30 miles an hour. It is therefore evident that when a body moves with variable velocity over a given space in a given time, the average velocity is equal to the uniform velocity with which any body. would pass over the same space in the same time. 28. Acceleration.-Bodies in nature do not move with uniform velocity, but motion with change of velocity is common. The rate of change of velocity is called acceleration. "Rate of change" is measured by the amount of change which takes place in a unit of time. Hence acceleration is the change of velocity per second. In ordinary language, the word "acceleration" means an increase of velocity, and the word "retardation" is used to denote a decrease of velocity; but in the science of mechanics, acceleration is the change of velocity per second, or the rate of change of velocity, whether the change be an increase, a decrease, or a change of direction, for velocity is a certain speed in a definite direction. Increase of velocity per second may be called positive acceleration, decrease of velocity negative acceleration. A body is said to have uniform acceleration when it receives equal increments or decrements of velocity in equal times, however small these times may be. For example, if a body is found to have at the end of successive seconds a velocity of 6, 9, 12, 15, and 18 feet per second, it is being uniformly accelerated, for the change of velocity per second is an increase of three feet per second each second, or three feet per second per second. Acceleration, it will be noticed, is measured by the increase of velocity per second. An acceleration of ten means an increase of 10 feet per second each second, and since the element of time comes in twice, we usually express this by saying 10 feet per second per second. As already mentioned, a retardation of velocity |