MECHANICAL PHILOSOPHY. DYNAM ICS. [iootLKBiTiifo 



described in the raccessiTe equal portions of time are as 

 tin- numbers I, 3, 6, 7, A'C. 



3. By means of the two equations (1), any one of the 

 four quantities/, t, , , may be eliminated, and an 

 equation deduced, involving only the other three ; so 

 that any one of the four things force, time, velocity, 

 and space may be expressed in terms of any two of the 

 others : thus, as the simplest algebraic substitutions 

 show, we have the following equivalent expressions : 



2f 

 Expressions for t>, ft, V 2/, -y 







/2 2. 

 " ' /' V f 



/>' 



2* 



4. A uniform accelerating force is measured by twice 

 the space described from rest in one second. 



2 



For putting t 1, we have/ ;j- = 2 ; so that if we 



can only measure the space through which a body, acted 

 upon by a uniform accelerating force, moves from a state 

 of rest, in one second, we can correctly determine the 

 inn/arm effect of that force on the body : it r ill be 

 dooblt the space thus passed through. In other words, 

 this double space will be the constant increment of the 

 velocity at each succeeding second ; that is, the constant 

 acceleration of the velocity. 



It is found by experiment, that the attraction of the 

 earth upon bodies near its surface that is, the force of 

 gravity causes a body to fall from rest a distance of 

 10 1 feet in the first second of time. Consequently, the 

 force of terrestrial gravity which force is usually re- 

 presented by g is g 32 -2 feet : that is, this force, 

 continuously soliciting a falling body, will accelerate its 

 Telocity 32-2 feet every second. 



It may not be amiss to repeat here, before proceeding 

 to practical illustrations, that where there is uniform 

 velocity, in a straight line, there is no force. We should 

 be compelled to admit this, as a necessary consequence of 

 our valuation of force in dynamics : where there is no 

 acceleration of velocity, there can be no force urging the 

 body onwards in its path. 



The student must all along remember, that when we 

 speak of a force acting on a body, we always refer to the 

 mechanical condition of the body at the particular instant 

 that has brought it to where we find it. The impulse 

 that, acting on a body at rest, puts it in a state of 

 uniform motion, acts only during the instant of passing 

 from rest to rectilinear motion : it expires, as it were, in 

 the act ; so that at whatever point in the path described, 

 the moving body comes under our examination, we are 

 compelled to say that no force not even the force of 

 impulsion is acting upon it then. 



We shall now give a few examples connected with 

 accelerating force. 



K\AMIM.K< oF A'VKI.KIIATINC VOKCK.-1. A 



body moves from rest with a uniformly accelerated 



' Telocity, and after the lapue of 3 minutes 5 seconds, is 



f. 'nM to have passed over 400 feet : required the accele- 



:ig force. 



Here the time, namely, t sees. 3 mins. 6 sees. 186 



i seen., and the space described, namely, 400 feet, are 



ii to determine /. From thu formula J/t* we 



li.i 1 . 



Consequently the accelerating 

 increase the Telocity -0234 feet every second. 



I low far will a heavy body fall in four seconds I 



Using g for/, we have 1 yt', where g 32-2, and 

 1-4, /. i - 16-1 X 4 1 - 257-6. Hence the distance 



& In what time will a heavy body descend 400 foot 1 



- \ gt* :. 400 - 



t - 



400 



6seoonds. 



4. If a body be projected vertically upwards with a 

 velocity of 400 feet per second, how high will it ascend ( 



It will ascend to that height from which, if it were let 

 fall, it would acquire a velocity of 400 feet upon reaching 

 the earth ; therefore since 



-.^.-..-iS-* 8 ""*- 



5. Through what height must a body fall so that the 

 velocity acquired may bo equal to tliat height I 



= -20 -64 -4 feet 



I * I. 4 -1 1 



i= 4 , but s = v . .1 = 4- 



g g 



6. Through what height must a body fall to acquire a 

 velocity of 100 yards a second ? Ans. 1311-75 feet 



7. A body is projected upwards with a velocity of | 

 1500 feet per second : how high will it ascend ? Ans. 

 34938 feet 



8. What time will elapse, after projecting the body in 

 the last example, before it again reaches the earth? 

 Ans. 93 seconds. 



In the foregoing inquiries respecting the rectilinear 

 motion of bodies under the influence of an accelerating 

 force, we have supposed the force to move the body from 

 rest. We are now to consider the case in which the 

 force begins to act upon a body already moving in the 

 direction of that force, or receiving an impulse in that 

 direction. Suppose a body to begin its motion with a 

 velocity ', and, from the commencement, to bo acted 

 upon by a uniform accelerating force / ; then, as before, 

 v being the velocity acquired in t seconds, and s the space 

 passed through in that time, the acquired velocity will 

 evidently be made up of the original velocity ' and the 

 velocity ft communicated by the force / ; for this force 

 adds to the velocity / feet every second, or ft feet in 

 t seconds, so that in t seconds the acquired velocity is 

 v - * -If ft. 



Again, in virtue of the initial velocity if alone, the 

 body would pass over the space v't in t seconds ; and in 

 the same time / alone would cause it to pass over the 

 space ^ft* : hence the combination of these is the whole 

 space passed over ; that is, it is i = v't -j- J/t a ; so that 

 the formulae for finding the velocity and the space 

 described are 



(2) 



If the initial velocity ' oppose that communicated 

 by /, then v' is to be taken with a sign opposite to that 

 of /. 



From these two equations t may be easily eliminated, 

 and a third equation obtained involving only the remain- 

 ing quantities ; thus, squaring the first, we have 



2 = t)' + 2 v'ft +f* < = *'" + 2/ (v't + $ ft*). 

 Hence, by the second equation, 



t,s = '2 -f 2/s . . . (3) 



As before, when gravity is the force,/ is replaced by g. 



Ex. 1. A body is projected vertically upwards, with :i 

 velocity of 480 feet per second ; at what height will it 

 be at the end of three seconds 1 



Hero 1/, the velocity of projection, being 480 feet, this 

 ity continued uniform for 3 seconds would carry it 

 to the height of 1440 feet ; but gravity so counteracts its 

 upward motion as to draw it back through a spmv ..f 

 $,,( - \ x 32.2 X 3" - 144-1) f > , t : hence the height 

 to which the body is suffered to ascend is only 1440 

 144-9 - 1295-1 feet ; that is, - v't J gt* - 11.UV1 

 f,, t. 



2. From an elevated position a body is projected ver- 

 tically upwards with a velocity of 80 feet : required its 

 place at the end of 6 seconds. 



i - i/< J ? < - 80 X 6 i 32-2 X 6 - 480 

 079-6 - '.I:H;. 



1 1. 'iice, the place of the body is 99'G foot below the 

 place of projection. 



