ART. 112.-Velocity in Space. When space is considered there are three rectangular components, as for example, and the direction is fully specified by the three direction-cosines Ex. 1. While a train is moving with a velocity of 20 miles an hour alongside a station platform, the guard throws out a parcel with a horizontal velocity of 16.9 feet per second in a direction at right angles to the motion of the train. What will be the velocity of the parcel at the beginning? First of all we have to see that the two components are expressed in terms of the same unit, say, feet per second. 1 hour 3,600 seconds, = Hence the tangent of the angle between the direction of motion of the parcel and the direction of motion of the train is '577; ... the angle is 30°. Also, since the components are rectangular, √(88)2 + (16·9)2 feet resultant = second, Ex. 2. A ship sailing due N. at the rate of 7 knots an hour is carried to the E. by a tide current of 4 knots an hour. Find her real velocity over the ground in knots correct to two places of decimals. 1. A steamer goes 9.6 miles per hour in still water. How long will it take to run 10 miles up a stream and return, the velocity of the stream being 2 miles an hour? 2. A party row down a river in three hours, and up in seven, their rate in still water being 5 miles an hour. Required the distance, and the velocity of the water. 3. Suppose that a steam-tug travels 10 miles an hour in still water when alone, but draws a barge 4 miles an hour. It has to take a barge 10 miles up a stream which runs 1 mile an hour, and then to return without the barge. How long will it take for the journey? 4. A vessel makes two runs on a measured mile, one with the tide in a minutes, and the other against the tide in 6 minutes. Find the velocity of the vessel through the water and of the tide, supposing both to be uniform. 5. A particle receives simultaneously three velocities, viz., 60 feet per second N., 88 feet per second W. 30° S., and 60 feet per second E. 30° S. Give the magnitude and direction of the resultant velocity. 6. A ship sailing due north at the rate of 8 knots per hour is carried to the east by a tide current of 4 knots an hour. Find her real motion over the ground in knots per hour correct to two places of decimals. 7. A river one mile broad is running downwards at the rate of 4 miles an hour, and a steamer moving at the rate of 8 miles an hour wishes to go straight across. How long will the steamer take to perform the journey, and in what direction must she be steered? 8. A boat is rowed in the direction of right across a river with a velocity of 8 miles an hour. The river has a velocity of 2 miles an hour, and a breadth of 800 feet. Find how far the boat will be carried down by the time it reaches the opposite bank. 9. A ship is sailing ESE. at the rate of 10 knots an hour, and the wind seems to blow from the NW. with a velocity of 6 knots per hour. Find the true velocity of the wind. 10. If a steamer have a velocity of 14 knots an hour due west, and the wind blows with a velocity of 7 knots an hour from the north; what will be the apparent velocity of the wind to one on board the steamer? SECTION XIX.—ANGULAR VELOCITY. ART. 113.-Speed of Turning. When a rigid body rotates round an axis, each point in the body has a speed proportional to its perpendicular distance from the axis. Hence speed of rotation or turning is expressed in the form Larc per L radius per T. Speed of turning may also be expressed in the form n revolutions = T. The reciprocal idea is that of periodic time, 1/n T per revolution. ART. 114.-Angular Velocity and Moment of Velocity. When a point moves in a plane, the rate of change of direction of the line joining it with a fixed point in the plane is called its angular velocity with respect to that point. It is specified, like speed of turning, in the form. or @ radian = T, Larc per T L radius. = Consider the motion in one plane of a point P round a fixed Fig.13. M 2 point 0 (Fig. 13). The moment of the velocity of P round 0 is proportional to the velocity of P, and to the perpendicular OM from O upon the direction of the velocity. Let the velocity be v L along per T, and the perpendicular be pL perpendicular. Then the moment of velocity is pv Lalong per T by L perpendicular, or pv L perpendicular by L along per T, or pv L area per T. This is double the rate at which the radius-vector describes area, because a small sector traced out by the radius-vector is a triangle, not a parallelogram. EXAMPLES. Ex. 1. The speed of the periphery of a mill-wheel 12 feet in diameter is 6 feet per sec.; how many revolutions does the wheel make per minute? Ex. 2. A person inquiring the time of day is told that it is between V. and VI., and that the hour and minute hands of the clock are together. What o'clock is it? rev. by hour hand = hour, Ex. 3. Find the average angular speed of the extremity of the minute hand of a watch which is three quarters of an inch in length. 2 inch arc= inch radius by hour, Ex. 4. Find the number of revolutions per mile made by a wheel of 4 feet diameter. Ex. 5. Express in degrees and in circular measure the angle made by the hands of a clock at 3.35 o'clock. 30 degrees by hour hand = hour, .. degree by hour hand = minute, 35 min.; ... The minute hand is 4 × 30 degs. past III., |