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 f be the acceleration, u the initial velocity, v 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 formula will present some interesting consequences; the student will obtain an illustration of the interpretation ascribed in Algebra to the negative sign.

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 t = 7; we have =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

u2

1 u2 f 2 f 27; 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.

From (2) we have s=0 when ut·

2u

t=0 and when t= The value t=0 corresponds to the 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 which is equal to the time taken ƒ ƒ' ƒ'

2u И

in moving forwards from the starting point to the turning

point. Put t

=

2u

f

in (1), then we get v=u-2u = −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

It will be important to remember these two results:

the original velocity u is destroyed in the time, and the

43. The most important application of the preceding Article is to the case of gravity. If a body be projected

vertically upwards with a velocity u it rises for a time,

u2

reaches the height falls to the ground in the same

2g'

time as it took to rise, and strikes the ground with the velocity u downwards.

EXAMPLES. IV.

1. A stone is thrown vertically upwards with a velocity 3g: find at what times its height will be 4g, and find its velocity at these times.

2. A body is projected vertically upwards with a velocity which will carry it to a height 2g: find after what interval the body will be descending with the velocity g.

3. A body moves over 20 feet in the first second of time during which it is observed, over 84 feet during the third second, and over 148 feet during the fifth second: determine whether this is consistent with the supposition of uniform acceleration.

4. A particle uniformly accelerated describes 108 feet and 140 feet in the fifth and seventh seconds of its motion respectively find the initial velocity and the numerical measure of the acceleration.

5. A body starts with a certain velocity and is uniformly accelerated: shew that the space described in any time is equal to that which would be described in the same time with a uniform velocity equal to half the sum of the velocities at the beginuing and the end of the time.

6. A bullet shot upwards from a gun passes a certain point at the rate of 400 feet per second: find when the bullet will be at a point 1600 feet higher.

7. A body is dropped from a given height and at the same instant another is started upwards, and they meet half way: find the initial velocity of the latter body.

8. At the same instant one body is dropped from a given height, and another is started vertically upwards from the ground with just sufficient velocity to attain that height: compare the time they take before they meet with the time in which the first would have fallen to the ground.

9. A smooth plane is inclined at an angle of 30° to the horizon; a body is started up the plane with the velocity g: find the time it takes to describe a space g.

10. A smooth plane is inclined at an angle of 30° to the horizon; a body is started up the plane with the velocity 5g: find when it is distant 9g from the starting point.

11. A body is thrown vertically upwards, and the time between its leaving a given point and returning to it again is observed: find the initial velocity.

12. A particle is moving under the action of a uniform force, the acceleration of which is f; if u be the arithmetic mean of the first and last velocities in passing over any portion h of the path, and the velocity generated, shew that uv=fh.

13. Two small heavy rings capable of sliding along a smooth straight wire of given length inclined to the horizon are started from the two extremities of the wire each with the velocity due to their vertical distance: find the time after which they will meet, and shew that the space described by each is independent of the inclination of the wire.

14. A body begins to move with the velocity v, and at equal intervals of time an additional velocity u is communicated to it in the same direction: find the space described in n such intervals. Hence deduce the space described from rest under the action of a force constant in magnitude and direction.

V. Second Law of Motion. Motion, under the influence of a uniform force in a fixed direction, but not in a straight line. Projectiles.

44. We are still confining ourselves to the case of a uniform force in a fixed direction; but the body will now be supposed to start with a velocity which is not in the same direction as the force: it will appear that a body under such circumstances will not describe a straight line but a certain curve called a parabola.

It is necessary at this stage to introduce the Second Law of Motion.

45. Second Law of Motion. Change of motion is proportional to the acting force, and takes place in the direction of the straight line in which the force acts.

So long as we keep to the same force and the same body change of motion is measured by change of velocity; the law then asserts that any force will communicate velocity in the direction in which the force acts and it is implied that the amount and direction of the velocity so communicated does not depend on the amount and direction of the velocity which may have been already communicated to the body. We shall see hereafter that the law contains more than this.

For the reason explained in Art. 10 we ought to suppose the Second Law to relate to the motion of a particle.

46. In confirmation of the truth of the Second Law of Motion it is usual to adduce the following experiment: If a stone be dropped from the top of the mast of a ship in motion the stone will fall at the foot of the mast notwithstanding the motion of the ship. The stone does not fall in a straight line; it starts with a certain horizontal velocity, namely, the same as that of the ship, and gravity acts on it in a vertical direction. The fact that the stone