Lesson Problems on bodies moving on a circle
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<H2>Problems on bodies moving on a circle</H2> <H3>Problem 1</H3>Jenna and Nina are running laps around the outside of their house in the same direction. They both start at the front door at the same time. Jenna takes 18 seconds to run all the way around. Nina takes 20 seconds. If they maintain the same speed, how long will it be before they are both at the front door at the same time again? <B>Solution</B> <pre> When Nina will complete her first loop, Jenna will be in 2 seconds ahead and will complete {{{2/18}}} = {{{1/9}}} of her second loop. When Nina will complete her second loop, Jenna will win additional {{{1/9}}} of the loop and will be ahead just in {{{2/9}}} of the loop. Every time when Nina will complete her next 20-seconds loop, Janna will win additional {{{1/9}}} of the loop. Finally, when Nina will complete her 9-th loop (180 seconds), Jenna will complete her 10-th loop, Thus, in 180 seconds after start, they both will be at the front door at the same time. <B>Answer</B>. In 180 seconds after start it will happen at the first time that both girls be at the front door simultaneously. </pre> <H3>Problem 2</H3>3 clocks, hour hands are missing, have minute hands running fast. Clock P, Q, R gain 2, 6, 12 mins per hour respectively. They start at midday - pointing to 12. Find the number of hours later when all 3 hands show the same number of minutes. <B>Solution</B> <pre> 1. The readings difference between clocks Q and P is increasing at the rate of 4 min per hour. It means that their readings will coincide every 15 hours, when their minute hands are at the same positions "verically up". It happens every 15 hours, and there is no other time moments, when their positions coincide. 2. The readings difference between clocks R and P is increasing at the rate of 10 min per hour. It means that their readings will coincide every 6 hours, when their minute hands are at the same positions "verically up". It happens every 6 hours, and there is no other time moments, when their positions coincide. 3. The readings difference between clocks Q and R is increasing at the rate of 6 min per hour. It means that their readings will coincide every 10 hours, when their minute hands are at the same positions "verically up". It happens every 10 hours, and there is no other time moments, when their positions coincide. 4. It implies that the next and closest time moment when the readings of all 3 clock coincide comes in LCM (15,6,10) which is 30 hours. LCM is "Leatt Common Multiple). <U>Answer</U>. In 30 hours. </pre> <H3>Problem 3</H3>Two cars, A and B, are driving in the same direction on a circular track, making one loop after the other. Car A takes 5 minutes to drive one whole loop. Car B makes the same in 7 minutes. The cars started simultaneously from the point P on the road. When the cars A and B will be simultaneously at the point P next time? <B>Solution</B> <pre> Intuitively, it is clear that the cars A and B will be simultaneously at the point P next time in N seconds, where N is least common multiple of the numbers of 5 and 7, i.e. after 35 seconds. The car A will complete 7 loops at that time, while car B will make 5 loops. Mathematically, the condition that the cars A and B will be simultaneously at the point P after making their loops, is this equality 5n = 7m seconds, where n and m are natural numbers. Thus the number 5n, which is the same as 7m, is a common multiple of numbers 5 and 7. </pre> <H3>Problem 4</H3>Tracy and Kelly are running laps on the indoor track at steady speeds, but in opposite directions. They meet every 20 seconds. It takes Tracy 45 seconds to complete each lap. How many seconds does it take for each of Kelly's laps? <B>Solution</B> <pre> When Tracy and Kelly are running on the track in opposite directions, the distance they cover together from one meeting point to the next meeting point is exactly the length of the track D. <U>It is the key moment to understand solving this problem.</U> Therefore, if "t" and "k" are the rates of each girl/women, then their summary (combined) rate is t + k = {{{D/20}}} per second in terms of the length D, (1) where D is the length of the track. From the other side, we know that the Tracy's rate is t = {{{D/45}}} per second in terms of the length D. Therefore, from (1) we have k = {{{D/20 - t}}} = {{{D/20 - D/45}}} = {{{(9D)/180 - (4D)/180}}} = {{{(5D)/180}}} = {{{D/36}}}. Thus, Kelly's rate is k = {{{D/36}}} per second in terms of D. It means that Kelly will cover the entire distance D (one lap) in {{{D/((D/36))}}} = 36 seconds. <U>Answer</U>. Kelly's lap will take 36 seconds. In other words, Kelly's rate is {{{D/36}}}, while Tracy's rate is {{{D/45}}}. <U>Check</U>. Their combined rate when they are running in opposite directions is {{{D/36 + D/45}}} = {{{(5D)/180 + (4D)/180}}} = {{{(9D)/180}}} = {{{D/20}}}. Check ! </pre> <H3>Problem 5</H3>Two track-men are running on a circular race track 300 feet in circumference. Running in opposite directions, they meet every 10 seconds. Running in the same direction; the faster passes the slower every 50 seconds. Find their rates in feet per second. <B>Solution</B> <pre> Let their rates are "u" and "v" ft per second, where "u" stands for the faster track-man. Then the first equation is {{{300/10}}} = u + v. (1) This equation is for the case then they are running in opposite directions. (Paradoxically, moving in opposite directions means moving towards each other along the circumference in this case !) The left side says that they together cover 300 ft in 10 seconds. The right side is the rate of decreasing the distance between them measured along the circumference. (Thinking on this equation, keep in mind that they meet for every circle counting after the first meeting). So the first equation is u + v = 30. (1') The second equation is {{{50u - 50v}}} = 300. (2) This equation is for the case they are running in one direction, and the equation says that for the faster track-man the path from one meeting point to the next meeting point is in one full circumference longer than for the slower track-man. The equation (2) is equivalent to u - v = 6. (2') The equations (1') and (2') are the governing equations, and we can easily solve them by adding. Doing so, you get 2u = 30 + 6 ---> 2u = 36 ---> u = {{{36/2}}} = 18. So the faster track-man speed is 18 ft/s. Then the slower track-man speed is 30-18 = 12 ft/s. <U>Check</U>. {{{300/(18+12)}}} = {{{300/30}}} = 10 seconds. {{{300/(18-12)}}} = {{{300/6}}} = 50 seconds. <U>Answer</U>. 18 ft/s and 12 ft/s. </pre> My other lessons on <B>Travel and Distance</B> problems in this site are <TABLE> <TR> <TD> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Travel-and-Distance-problems.lesson>Travel and Distance problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Travel-and-Distance-problems-for-two-bodies-moving-toward-each-other.lesson>Travel and Distance problems for two bodies moving in opposite directions</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Typical-catching-up-Travel-and-Distance-problems.lesson>Travel and Distance problems for two bodies moving in the same direction (catching up)</A> - <A HREF=http://www.algebra.com/algebra/homework/NumericFractions/Using-fractions-to-solve-Travel-problems.lesson>Using fractions to solve Travel problems</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Wind-and-Current-problems.lesson>Wind and Current problems</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/More-problems-on-upstream-and-downstream-round-trips.lesson>More problems on upstream and downstream round trips</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Wind-and-Current-problems-solvable-by-quadratic-equations.lesson>Wind and Current problems solvable by quadratic equations</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Unpowered-raft-moving-downstream-along-a-river.lesson>Unpowered raft floating downstream along a river</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-problems-from-the-archive-on-a-boat-floating-Upstream-and-Downstream.lesson>Selected problems from the archive on the boat floating Upstream and Downstream</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-problems-from-the-archive-on-a-plane-flying-with-and-against-the-wind.lesson>Selected problems from the archive on a plane flying with and against the wind</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-Travel-and-Distance-problems-from-the-archive.lesson>Selected Travel and Distance problems from the archive</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Had-a-car-move-faster-it-would-arrive-quicker.lesson>Had a car move faster it would arrive sooner</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/How-far-do-you-live-from-school.lesson>How far do you live from school?</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/One-unusual-Travel-problem.lesson>One unusual Travel problem</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Another-unusual-Travel-problem.lesson>Another unusual Travel problem (Arnold's problem on two walking old women)</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Travel-problem-on-a-messenger-moving-back-and-forth-along-the-marching-army-column.lesson>Travel problem on a messenger moving back and forth along the marching army's column</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/A-person-walking-along-the-street-and-the-buses-traveling-in-the-same-and-opposite-directions.lesson>A person walking along the street and buses traveling in the same and opposite directions</A> </TD> <TD> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Calculating-an-average-speed.lesson>Calculating an average speed: a train going from A to B and back</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/One-more-problem-on-calculating-an-average-speed.lesson>One more problem on calculating an average speed</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Clock-problems.lesson>Clock problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Advanced-clock-problems.lesson>Advanced clock problems</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-a-telegraph-post-and-passing-a-bridge.lesson>A train passing a telegraph post and passing a bridge</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-a-platform.lesson>A train passing a platform</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-a-tunnel.lesson>A train passing through a tunnel</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-light-rail-train-passing-a-walking-person.lesson>A light-rail train passing a walking person</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-another-train.lesson>A train passing another train</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/A-man-crossing-a-bridge-when-a-train-comes-from-behind.lesson>A man crossing a bridge and a train coming from behind</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/27-A-problem-on-a-rower-going-on-a-river-who-missed-the-bottle-of-whiskey-under-a-bridge.lesson>A rower going on a river who missed the bottle of whiskey under a bridge</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Non-traditional-Travel-and-Distance-problems.lesson>Non-traditional Travel and Distance problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/The-distance-covered-by-a-free-falling-body-during-last-second-of-the-fall.lesson>The distance covered by a free falling body during last second of the fall</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/The-Doppler-shift.lesson>The Doppler Shift</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Entertainment-Travel-and-Distance-problems.lesson>Entertainment Travel and Distance problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/OVERVIEW-of-lessons-on-Travel-and-Distance.lesson>OVERVIEW of lessons on Travel and Distance</A> </TD> </TR> </TABLE> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I.
