CHAPTER IV. ON PROJECTILES. IN the last chapter the motion of a body projected vertically upwards or downwards, under the action of gravity, was considered; the whole motion taking place in the vertical line through the point of projection. When the direction is any other than vertical, the path of the body is an arc of the curve called the parabola. By the second law of motion, gravity produces its full effect independent of the motion of projection: and we may consider the latter as compounded of a horizontal and vertical motion. The latter of these only can be affected by the action of gravity on the body. 14. PROP. To determine the path of a body projected in a given direction, with a given velocity, under the action of gravity. If we draw AVx' a vertical line, and taking AV = PT, complete the parallelogram ATPV, we have AV = x', PV = y', the oblique co-ordinates of the point P. Let also h be the height from which the body must fall to acquire the velocity v, or h= v2 2g' we have from (1) y=4hx' which, as seen in treatises on conic sections, is the equation to a parabola whose axis is parallel to Ax', and therefore vertical, Ay' a tangent at the point A, and h the distance SA of the focus, and also of the directrix from A. With these data the parabola to represent the path of the body can be described. 15. PROP. To find the equation to the path of a projectile when referred to axes of co-ordinates which are horizontal and vertical. the equation required; or substituting, as in the last Prop., DEFINITIONS. The horizontal range of a projectile is the distance AB from the point of projection to the point where it strikes the horizontal plane in its descent. The time of flight is the time it takes in describing APB. 16. PROP. To find the time of flight of a projectile on a horizontal plane. We have generally, as in the last proposition, y=vtsin. a-gt2 and if we put y = 0, or vtsin. a-gt-0, the result will apply to the points A and B only; and the values of t are which is therefore the time of flight required. We should have arrived at this result by the same method as in Article 13, by putting for u, the vertical component of the velocity of projection, v sin. a. 17. PROP. To find the range of a projectile on a horizontal plane. The result, as before, applies to the points A and B, and This value of AB varies with the angle of elevation ɑ, and is greatest when sin. 2a = 1, or a 45°, v remaining the same. = L We see that the horizontal range is the space which would be described with the uniform horizontal velocity v cos. a, in the 2v sin. a time of flight 18. PROP. To find the greatest height which a projectile attains. The greatest altitude is evidently the value of y at the middle point of the path above a horizontal plane, or when the time is v sin. a one-half of the time of flight, or t = 9 greatest altitude the projectile attains is that to which it would rise by the vertical component v sin. a of the velocity of projection. 19. PROP. To shew that the velocity at each point of the parabolic path of a projectile is that which would be acquired in falling directly from the directrix. In Article 14 it was shewn, that if h be the height due to the velocity of projection, it is also the distance of the point of projection from the directrix of the parabola described. Now, if any point in the parabola were taken for the point of projection, and a body were projected from it with the same velocity and direction which it has in the parabola, it would describe the same parabola; and therefore what holds for the point of projection holds also for all other points of the path. 20. PROP. To find the point where a projectile will strike an inclined plane through the point of projection, and its distance, or the range on the inclined plane. Let y=x tan. ẞ be the equation of the line AC, which is the intersection of the inclined plane with the vertical plane in which the body is projected. Combining this with the equation of the path of the projectile in Article 15, namely, y A y=x tan. a. 4h cos.2 a we have the co-ordinates of the point C cos. a. sin. (α- -B) C B x = 4h cos. B and the distance AC = √x2+ y2 = x sec. B |