Kinematics of Three Dimensions

KINEMATICS OF THREE DIMENSIONS Similarly, the speed v with which the ship is travelling north will be given by (8) The actual direction of motion, when the ship is at P, is along the tangent at P to the path QPR; the speed along the path has, at this instant, some definite value S. We say that the ship has a velocity S in the direction of the tangent. Thus velocity is a quantity which has the nature of a speed, but which also possesses direction; in equations (7) and (8), we may say that u is the east ward and v the northward velocity of the ship.

Composition and Resolution of Velocities.-6.

We have seen that the actual velocity S of the ship at P may be regarded as a combination of the two velocities a and v: when u and v are both known, it must evidently be determinate, both in magnitude and direction. To find the required relations between S, u and v, we have only to imagine that the ship, on reaching P, maintains its velocity unchanged. The path then becomes a straight line PT (fig. 3), and in an interval t the ship will go to T, where as in (i). Also, since u is the eastward velocity, and LH (or PM) the eastward distance covered in the interval t, we have hence, dividing corresponding sides of (9) and (I o), we deduce that The Parallelogram Law.—Thus (fig. 4) the component velocities v and the resultant or total velocity S can be represented both in direction and magnitude by the sides and diagonal, respectively, of a parallelogram. In this particular instance the parallelogram is a rectangle; but the same result can be established when (as in fig. 5) the directions of u and v (i.e., the directions of the coordi nates x and y) are not perpendicular. The total velocity S may be regarded as made up. of two velocities, u, v, represented in direc tion and magnitude by PM and PN, the sides of any parallelogram of which PT is a diagonal.

The Vector

Two coordinates x, y suffice to define the position of a ship, because this is (practically speaking) a body moving in one plane. To define the position of an aeroplane, we require a third coordinate; viz., the height z above some stand ard level. Corresponding to (7) and (8), we have the expression for the vertical component of acceleration.

The parallelogram law has an obvious extension in three dimen sions. If OP, the diagonal of a parallelopiped (fig. 6), represents a velocity or acceleration in di rection and magnitude, this may be resolved into components rep resented in direction and magni tude by OA, OB, OC, the sides of the parallelopiped. Speaking gen erally, we say that velocities and accelerations may be resolved and compounded according to the vector law.

Motion of a Projectile in Vacuo.-9.

A simple example will serve to explain the application of these kinematical principles. We may assume (as a deduction from experiment) that a body moving in vacuo near the surface of the earth has a constant total acceleration g directed vertically downwards : making this assump tion, we proceed to calculate its path, or trajectory.

To define the position of the moving body (or projectile), we take axes Ox, Oy, Oz, fixed in relation to the earth. We take Oz to be directed vertically upwards; i.e., in the line of the resultant acceleration. Then the acceleration has no component along Ox or Oy, and we have from (12) and (14), Now let Oy be taken horizontal and perpendicular, at some definite instant to the direction of motion; then, at this instant, there is no component velocity in the direction Oy; i.e., v=o. Hence, according to the second of equations (15), v will be zero always; i.e., the trajectory lies wholly in the plane z0x.

The problem is now two-dimensional. If are the initial components of velocity (viz., at time t=o), we have from (7) and (i5), cal standpoint the notion of force is unnecessary; but it is con venient, since it enables us (in theory) to confine attention to a single body. Thus, in the problem just discussed, we are really concerned with the relative motion of two bodies, the projectile and the earth; but we can confine attention to the projectile, once we have postulated that its interaction with the earth is equivalent, so far as its own motion is concerned, to the imposition of a force which gives it a constant downward acceleration g.