IV. Motion in a straight line under the influence of a uniform force, with given initial velocity. 35. In the preceding Chapter we confined ourselves to the case in which the body was supposed to have no velocity before the force began to operate; this supposition is usually expressed by saying that the body has no initial velocity. We shall now suppose that the body has an initial velocity, the direction of which coincides with the straight line in which the force acts. 36. A body starts with the velocity u, and is acted on by a uniform force in the direction of this velocity during the time t: iff be the acceleration, and v the velocity of the body at the time t, then v=u+ft. For, by the definition of uniform force, in each unit of time the velocity f is communicated to the body; and therefore in t units of time the velocity ft is communicated: therefore at the end of the time t the velocity is u+ft. 37. A body starts with the velocity u, and is acted on by a uniform force in the direction of the velocity during the time t: if f be the acceleration, and s the space described in the time t, then s= ut + ft2. 1 2 Let the whole time t be divided into n equal intervals; denote each interval by r, so that nr=t. Then the velocity of the body at the end of the times from starting is, by Art. 36, respectively u+fr, u+2ft, u+3ft, ...... u+(n−1) fr, u+nfr. Let $1 denote the space which the body would describe if it moved during each interval with the velocity which it has at the beginning of the interval; and let s2 denote T the space which the body would describe if it moved during each interval with the velocity which it has at the end of the interval. T Then 81 = UT + {U+fr}T+ {u+2fT} T+......+ {u + (n-1) fr} T, 82={U+fr}T+{u+2fr} r+ ... that is, ...+{u+(n-1) fr} + {u+nfr}T; 81 = unr+fr2{1+2+3+......+ (n-1)}, Hence, by the theory of Arithmetical Progression in Algebra, we have ft2 = ut+' 2 82 = Un++ƒ+2n (n + 1) = ut + m2 (1 + 1). 2 Now s, the space actually described, must lie between 1 s, and s; but by making n large enough we can make n as small as we please; so that we can make s1 and s, 1 differ from ut+ft by less than any assigned quantity. 1 2 Hence s=ut+ft2. 2 38. The result just obtained has been deduced by an independent investigation founded on first principles; if we are allowed to assume the result obtained in Art. 21 we may put the investigation more briefly as follows: If the body at a certain instant is moving with a certain velocity, its subsequent motion will be the same, however we suppose that velocity to have been acquired. Let us suppose that the velocity u was generated by the action of the force, of which the acceleration is f, during the time t'; and let the body have moved from rest through the spaces' during this time. Then we have, by Art. 21, u =ft', 39. The result of Art. 37 is sometimes obtained in the following way: If no force acted on the body the space described in the time t would be ut, by Art. 5. If there were no initial velocity the space described in the time t under the in 1 fluence of the force would be ft. Now if the body start with the velocity u, and be also acted on by the force, the space actually described must be the sum of these two spaces; because by the nature of uniform force the velocity at any instant is exactly the sum of what it would be in the two supposed cases. 40. Hence we have the following results when a body starts with a given velocity and is acted on by a uniform force in the direction of this velocity. Let f be the acceleration, u the initial velocity, the velocity at the end of the time t, and s the space described; then thus v=u+ft From (1) and (2) we have v2 = u2+2uft+ƒf 2t2=u2+2ƒ ( ut + ft2 v2=u2+2fs. ..(1). (2). (3). 41. The student must observe that during the motion which we consider in Art. 37 the only force acting is that of which the acceleration is f. The body starts with the velocity u, and this must have been generated by some force, which may have been sudden, as a blow or an explosion is usually considered to be, or may have been gradual like the force of gravity. But we are only concerned with what takes place after this velocity u has been generated, and so during the motion which we consider no force acts except that of which the acceleration is f. 42. Hitherto we have supposed the direction of the force to be the same as that of the initial velocity; we will now consider the case in which the direction of the force is opposite to that of the initial velocity. It will be sufficient to state the results, which can be obtained as in Arts. 36, 37, 38, and 40. Let be the acceleration, u the initial velocity, the velocity at the end of the time t, and s the space described, the force and the initial velocity being in opposite directions; then These formulae will present some interesting consequences; the student will obtain an illustration of the interpretation ascribed in Algebra to the negative sign. น As long as ft is less than u we see from (1) that v is positive, so that the body is moving in the direction in which it started. When ft-u=0, that is when t: we have v=0, so that the body is for an instant at rest. When t is greater than the value of u is negative; f that is, the body is moving in the direction opposite to that in which it started. Thus we see that the body continues to move in the direction in which it started, until by the operation of the force, which acts in the opposite direction, all its velocity is destroyed; after this the force generates a new velocity in the body in the direction of the force, that is, in the direction opposite to that of the original velocity. From (2) when t= we have s= f u2 1 u2 u2 f = 2; this gives the whole space described by the body while moving in the direction in which it started. This value of s may also be obtained from (3) by putting v=0; for then we have u2-2fs=0. 1 From (2) we have s=0 when ut- ft2=0, that is when 2u 2 t=0 and when t= The value t=0 corresponds to the f instant of starting; the other value of t must correspond to the instant when the body in its backward course reaches the starting point again. Thus the time taken in moving backwards from the turning point to the starting point is or J' which is equal to the time taken 2u и in moving forwards from the starting point to the turning Qu point. Put t= in (1), then we get v=u-2u= f u; so that at this instant the velocity of the body is the same numerically as it was at starting, but in the opposite 2u direction. When t is greater than the value of s be f comes negative, indicating that the body is now on the side of the starting point opposite to that on which it was while t changed from 0 to 2u f It will be important to remember these two results; the original velocity u is destroyed in the time, น , and the |