If the scale overtakes the vernier after say 7 divisions, then the gain by the scale is 7 inch; that is, the difference between the approximate and true reading is 7 inch. When the equivalence for the vernier is then 1.1 inch of vernier division, = 1 inch gain by vernier division. = In this case the coincidence has to be sought for by going backwards instead of forwards. A vernier constructed according to the former plan is called a sextant-vernier, while one constructed according to the latter plan is called a barometer-vernier. EXAMPLES. Ex. 1. Two passenger trains having equal speeds, and consisting, the one of 12 carriages, the other of 14, are observed to take 10 seconds to pass one another. What is the speed, estimating the length of a carriage at 23 feet? Let the speed of either train be Ex. 2. In running a race one mile long A beats B by 100 yards, and B beats C by 90 yards; by how many will A beat C? 1760 yards by A 1660 yards by B, = 1760 yards by B = 1670 yards by C, 1760 yards by A; 166 × 1670 yards by C, 176 i.e., 15755 yards by C'; 5 44 1760 – 1575 yards by which A beats C, i.e., 184 yards by which A beats C. Ex. 3. A racecourse is 3,000 ft. long; A gives B a start of 50 ft. and loses the race by a certain number of seconds; if the course had been 6,000 ft. long, and they had both kept up the same speed as in the actual race, A would have won by the same number of seconds. Compare A's speed with B's. Similarly in the other case the time by which A wins is Ex. 4. A man walks, at three miles an hour, along a tram-line, and during his walk he is overtaken by 6 tram-cars. If the cars start simultaneously at equal intervals of time, from both ends, and travel at the rate of 5 miles an hour, determine the number of tram-cars he should have met in the same time. 1. A traveller starts from A towards B at 12 o'clock, and another starts at the same time from B towards A. They meet at 2 o'clock at 24 miles from A, and the one arrives at A while the other is still 20 miles from B. What is the distance from A to B? 2. A steamer leaves Liverpool for New York, and a vessel leaves New York for Liverpool at the same time; they meet, and when the steamer reaches New York the vessel has as far to go as the steamer had when they met. If Liverpool be 3,000 miles from New York, how far out from Liverpool was the steamer when they met? 3. A starts to walk from Edinburgh to Glasgow, and at the same time B starts to walk from Glasgow to Edinburgh. A reaches Glasgow 9 hours after meeting B, and B reaches Edinburgh 6 hours 15 minutes after meeting A. Find in what time each has performed the journey. 4. A, who walks at the speed of 3 miles per hour, starts 18 minutes before B; at what speed must B walk to overtake A at the ninth milestone? 5. A messenger starts on an errand at the rate of 4 miles an hour; another is sent 13 hours after to overtake him; the latter walks at the rate of 4 miles an hour. When and where will he overtake him? 6. A passenger train going 41 miles an hour, and 431 feet long, overtakes a goods train on a parallel line of rails. The goods train is going 28 miles an hour, and is 713 feet long. How long does the passenger train take to pass the other? 7. A and B run a race a mile long, and A beats B by 100 yards; A then runs with C, and beats him by 200 yards; finally B runs the course with C. By how much does B beat C? 8. A and B run a race; B has 50 yards start, but A runs 20 yards while B runs 19. What must be the length of the course that A may come in a yard ahead of B? 9. In a 100 yards race A beats B by 5 yards and C by 10 yards. By how many yards does B beat C? 10. In a 100 yards race A can beat B by 10 yards; B in the same distance beats C by 10 yards. By how many will A beat C? 11. In a mile race A can beat B by 17 yards or 25 seconds. Find A's time over the course. 12. In a mile race A gives B 50 yards; B passes the winning post 5 minutes after the start; A passes it 5 seconds later. Which would win in an even race, and by what distance? 13. In a mile race between A and B, whose relative speeds are as 4 to 3, B had the start by 3 minutes, but was beaten by 80 yards. Required the speed of each in yards per minute. 14. A starts from a railway station, walking at the rate of 5 miles an hour; at the end of an hour B starts walking 4 miles an hour; at the end of another hour a train starts and passes A 25 minutes after it passes B. Find the speed of the train. 15. A tourist, having remained behind his companions, wishes to rejoin them on the following day. He knows they are 5 miles ahead, will start in the morning at 8 o'clock, and will walk at the rate of 3 miles an hour. When must he start in order to overtake them at 1 o'clock P.M., walking at the rate of 4 miles an hour, and resting once for half an hour on the road? 16. One man setting out from A travels towards B at the rate of 6 miles per hour; 2 hours afterwards a second man starts from A, and going 10 miles per hour reaches B 4 hours before the first man. Find the distance between A and B. 17. Two men start from a town on the same road-the first on foot, walking 18 miles in 7 hours; the second on horseback 53 hours later, walking 36 miles in 5 hours. Find the time in which the second gets (1) half as far, and (2) twice as far as the first. 18. Suppose that cars move on a tram-route at the average rate of 6 miles per hour, and are despatched from either end at intervals of 5 minutes, and that a man walks along the route at the rate of 4 miles an hour. How many cars per hour will meet him, and how many cars per hour will overtake him? 19. A man walking at 4 miles an hour along a tram-route observes that in the course of an hour he meets 20 cars, and is overtaken by 4. What is the average speed of the cars, and what is the average distance between two successive cars? 20. Show how to construct a vernier to make barometer readings to '002 of an inch, when the divisions on the scale are twentieths of an inch. 21. The circumference of the limb of an angular instrument is divided into 1,080 parts; show how to construct a vernier which will read to a minute. SECTION XVIII. -VELOCITY. ART. 111.-Velocity in one Plane. Velocity, being a vector opposite Fig.12. Lalong L expressed by (Fig. 12) rate, can be resolved into components in the same manner as a vector. Suppose that our attention is restricted to one plane. The equivalence between the full velocity and its components is v L along per Tv, L adj. per T+ L opp. per T. From this complete equivalence certain partial equivalences may be derived, as in Art. 25, This last, when the components are at right angles to one another, is the cosine of the direction of the velocity. Similarly, under the same condition, V2L opp. = L along ช v2L opp. per T = v1 L adj. per T; VL opp. = L adj. ; which, when the components are rectangular, is the tangent of the direction of the velocity. The relation between the numbers is the same as that for the resultant and components of a simple vector. When the com ponents are inclined at an angle 0°, v2 = v2 + v2 + 2v1v2 cos 0. v2 = v2 + v2. When is 90 this becomes |