ne 



MECHANICAL PHILOSOPHY. DYNAMICS. [PATH OF PROJECTILES. 



4. If a body be projected up an inclined plane whose 

 length u ton time* iU height, with a velocity of 30 feet, 

 in what time will the Telocity be destroyed, and the body 

 begin to roll down 1 



The time U obviouily the tame u would be required 



luce a Telocity of 30 feet in a body falling from 



rut down the lame plane : hence, substituting the 



acceleration? . for /in the expression for ( (page 730), 



we hare 



rl 30 X 10 300 



In the foregoing investigation!), the student will 

 observe that no allowance U made for the friction of 

 bodies rolling down inclined planes ; nor, whether bodies 

 fall vertically, or obliquely as in this article, is the 

 resistance of the air token into account These hin- 

 drances to free motion can be estimated only by practical 

 and experimental researches, under various circum- 

 stances and conditions. The resistance of the air 

 cannot be provided against, as is obvious ; but in pieces 

 of delicate machinery, many ingenious contrivances are 

 resorted to, to diminish friction, and to render the 

 departure from rigid mathematical theory as trifling as 

 possible. The subject of friction will come under con- 

 sideration in the section on APPLIED MECHANICS. 



THE PARALLELOGRAM OF VELOCITIES. The 

 forces considered in Dynamics are influences of the same 

 kind as those considered in Statics ; they merely exhibit 

 their effects in a different manner. What in Statics 

 produces pressure, would in Dynamicsthat is, if the 

 thing pressed were removed produce motion : the 

 effects in both cases being proportional to the causes : 

 a double pressure from the same body, like a double 

 acceleration of velocity, implies a double force. 



It may, however, be proper here to anticipate a dif- 

 ficulty which the student may possibly feel : he may 

 perhaps reason thus : " A weight of ten pounds pro- 

 duces a pressure ten times that of one pound, yet the 

 one pound weight, if let fall, is accelerated just as much 

 as the ten-pound weight ; whereas, from what is here 

 said, it would seem that the heavier weight ought to be 

 accelerated ten times as much as the lighter, seeing that 

 its pressure is ten times as great" 



But this apparent difficulty can arise only from a 

 wrong conception of what is stated above : we are not 

 comparing the pressure of different bodies under the 

 same force acting upon all their particles, but the pres- 

 sure of the tame body under different acting forces : for 

 example, if the same body were acted upon, now by the 

 force of gravity, and hereafter if the same body, or an 

 equal body, were acted upon by a force only half that of 

 gravity, then, as stated above, the effect in the former 

 case, whether that effect be pressure or acceleration, 

 would be double the like effect in the latter case. If it 

 were otherwise, than the statical measure of force (pres- 

 sure) and the dynamical measure (acceleration of velo- 

 city) could not each of them be a correct measure or 

 expression of the intensity of what we call force. 



it has been fully proved in 

 STATIC* (page 694), that if two 

 forces P and Q act at A, and 

 A It, A C represent their direc- 

 tions and intensities respec- 

 tively, then will A D represent 

 in direction and intensity their j 

 united effect 



The statical effect remains the same however long 

 the forces or pressures act for time is no element of 

 consideration in Statics. But the dynamical effect is 

 that at the instant the body acted on by the forces U at 

 A, the continuous action of the forces, afterward*, not 

 entering into consideration. 



The parallelogram O f vtlocitia is independent of the 

 parallelogram of force*. It will be remembered that the 

 velocity of a body at any [K.int of iU path U the space 

 it icvuW pass over in a unit of time, provided its rate or 



peed at that point were uniformly continued during the 

 unit of time. 



Suppose the body, when at A, were animated with a 

 velocity that would alone carry it uniformly along A B 

 to B, and with another velocity that would alone carry 

 it uniformly along A C to C in the same time : these 

 velocities may be considered as communicated by two 

 simultaneous impulses in the directions A B, A C : the 

 thing to be shown is, that the body would be carried 

 uniformly along A D, and would arrive at D in the time 

 spoken of. 



For while the body is moving uniformly from A to B, 

 conceive that the line A B, with the moving body on it, 

 is carried parallel to itself, and with the second uniform 

 velocity, up to C D : by the hypothesis, it will have 

 arrived at C D in the same time that the body will have 

 arrived at the extremity of the moving line : conse- 

 quently, at the end of that time the body will be found 

 at D. And that it will have arrived there with a uni- 

 form motion along the diagonal A D, will appear from 

 considering, that if any point of its path were out of 

 that diagonal, the uniform relation of the component 

 velocities, namely A B : A C, would there be destroyed. 

 That the diagonal is described with a uniform velocity 

 or that equal portions of it are passed over in equal 

 portions of the time, is plain, because the uniformly 

 moving line A B passes over equal portions of the diago- 

 nal in equal portions of the time. 



Since, in the preceding figure, trigonometry gives for 

 AD the expressi.. n 



AD>= AB a + BD 5 2AB-BDc>s. ABD, 



and since cos. A B D = + cos. B A C, if the directions 



of the component velocities v, v' make an angle a with 



each other, then the resultant velocity V will be 



V* = + p 78 + 2rr / cos. a 



From attending to the former part of the preceding 

 examination, it will be seen that, however irregular the 

 motion that would carry A to B, and however irregular 

 the motion that would carry it to C, in the same time, 

 the resultant of the two motions would necessarily carry 

 it to D in that time, although not by the path A D, 

 except the two component motions, during simultaneous 

 portions of the time, are always as A B to A C. This 

 truth will be of considerable importance in the next 

 article on projectiles. 



ON THE MOTION OF PROJECTILES. If a body 

 be projected obliquely upward or downward, the attrac- 

 tive force of gravity will cause it to take a curvilinear 

 path. If the velocity of projection be considerable, and 

 the body projected present much surface to the atmo- 

 sphere, the resistance of the air will greatly modify the 

 form of the curved path, and the range of the projectile ; 

 and it is no easy matter to determine one or the other. 

 But if the body be supposed to move free from such 

 obstmction (as in vacuum), all the circumstances con- 

 nected with the flight of a projectile can bo readily 

 ascertained, as also, with a close decree of approxima- 

 tion, even in actual practice, if the velocity of projection 

 be but small. 



Supposing atmospheric resistance to be removed, the 

 path of the projectile will always be the conic section 

 called a parabola (see PRACTICAL OBOMETBT, page 610), 

 as we now proceed to show. 



PATH OF PROJECTILE A PARABOLA. Let 

 A B (Fig. 143) be the direction in which a body is pro- 

 jected from the point A, with a velocity v. If no other 

 motion were impressed upon the body, it would move 

 uniformly along A B with the original velocity v, and in 

 t seconds would arrive at B, supposing A B=t. But 

 as gravity acts on the body from the commencement, 

 this force alone, in t seconds, would draw the body 

 down, along AC, to C, supposing AC fat* (page 736). 



Consequently, completing the parallelogram B C, the 

 body in t seconds is found at P, as proved at the close 

 of hut article. Now 



CP-AB-vf, and AC= J-^ 1 

 CP 1 2e' 2t>" 



Ac-T" CP ?' AO 



