Two cyclists, 120 miles apart, approach each other, each pedaling at 10 miles per hour. A fly starts at one cyclist and flies back and forth between the cyclists at 15 miles per hour. When the cyclists come together (and squash the fly between them), how far has the fly flown?  I know that I am supposed to find some sort of series here, but I don't really know how to approach this. How do I find the terms?
74358
The Problem:  Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 50km apart. A fly starts at the front of one train and flies at 75 kph to the front of the other; when it gets there, it turns around and flies back towards the first. It continues flying back and forth til the two trains meet and it gets squashed (the least of our worries, perhaps).  How far did the fly travel before it got squashed?  Attempt at a solution:  I can do this by summing the infinite series of the fly's distance for each leg. I get an answer of 37.5 km: but that's so nice! There must be a more intuitive way...is there?
34987
Trying to study for my mid-term, but I'm having slight difficulties understanding what I'm supposed to do in this one problem:  A batter starts running towards first base at a constant speed of 6 m/s. The distance between each adjacent plate is 27.5 m. After running for 20 m, how fast is he approaching second base? At the same moment, how fast is he running away from third base? (see image below)    This is what I have so far:   Let $d$ be the distance the batter has run thus far The distance between the batter and first base is 7.5 m The distance between the batter and second base is $\sqrt {27.5^2 + (27.5-d)^2}\ $, or approx. 28.5044 m when $d = 20$ The distance between the batter and third base is $\sqrt {27.5^2 + d^2}\ $, or approx. 34.0037 m when $d = 20$   No need to hand feed me the answer, I'd just like a bit of insight on how to solve the problem.
17088
Everyone knows the story about countersteering. For those who don't, I will explain it below and after the explanation I will ask my question.  You can watch this short video as a beginning:   .  This an excellent graphic demonstrating what is happening:    The key is that moving your butt moves the CG ($y$ and $x$ are the location in the image.) Basically what has to happen is for the moments created from the weight ($mg$) and the centrifugal force $\left(\frac{mv^2}{r}\right)$ to cancel out. (A moment is a force times a distance, in this case the distance to the contact patch in the direction parallel to the force - $mg\times y$ for weight, $\frac{mv^2}{r}\times x$ for centrifugal force.). You can express $x$ and $y$ with sine and cosine times the distance from the contact point to CG. Then the bike goes around the corner and sparks fly from your titanium knee sliders. You can speed up and lean over (or drop your butt) until the centrifugal force equals the maximum force that the tires/road combination can handle. This is of course assuming that you're not doing anything else like braking or accelerating which takes away from the grip available for cornering. If you want the equations to play with: $\frac{mv^2}{r}\times y = mg \times x , m = mass$, cancels out! It doesn't matter how fat your ass is for computing the speed/turn radius, just where it is and the ratio of ass fat to bike fat (it does matter for maximum speed though) -> $v =$ velocity -> $r =$ radius of turn -> $y =$ vertical distance of cg from center of rotation (contact patch) -> $g = $ acceleration due to gravity (constant $9.81$ $m/s^2$) -> $x = $ horizontal distance of CG from center of rotation The two forces are trying to rotate the bike in opposite directions which is why the are on opposite sides of the equation. so: $v = \sqrt{\frac{rgx}{2y}}$ OR $r = \frac{2yv^2}{gx}$ You can see that as $y$ drops or $x$ gets bigger (lean over more or hang off more for both, basically) either $r$ has to go down (tighter turn) or $v$ has to go up!  Now comes my question. In ALL explanations (including my own) the centripetal force in the picture (free body diagram) is pointing outwards. But this doesn't make sense, because the centripetal force should point inwards. Then you get the problem that BOTH torques try to turn you to the inside. In practice we can obviously see that this doesn't happen. So you would say that in this case the centripetal force (better: centripetal torque) is pointing outwards. How is this possible? I have always learned that the centripetal force should point to the centre of the circle. Please help !!!
60996
I am looking at buying a pedal/shoe system as the clips and straps I am using are hurting my feet again.  It's been a long time since I have bought a road pedal / shoe system, and I have forgotten the differences between SPDs, Look pedals and all the others.    Can someone give a short comparison of each pedal system with pros and cons of each? My usage is/will be mainly city street cycling and then road cycling (recreation, not racing).  I'd prefer if people would give the pros and cons of each and not their opinion on the "pedal system war".  :) However, "I use them because I find XYZ and it helps in ABC ways" is very useful.
79271
Problem:     Two people are to arrive at the same location some time between 3pm   and 4pm. They both arrive at random times within the hour and only   stay for ten minutes. Since they only go one time, what is the   probability that they will meet during that hour?   I have found the correct answer to this problem here (the second formula P2):      $$ P2 = 1 - \frac{(1-w_1)^2 - (1-w_2)^2}{2} = w_1 + w_2 - \frac{w_1^2}{2} - \frac{w_2^2}{2} $$  Plugging in ($1/6$) for both $w_1$ and $w_2$ is $11/36$ which is the correct answer.  But I don't understand how they came to that conclusion and what formula they derived it from. Any help in explaining this would be much appreciated!
84317
I am going to buy a road bike. I am considering a .    I used the  to find the size of bike that fits me.    Inseam:             80.5   Trunk:              59.5   Forearm:            36   Arm:                67.2   Thigh:              57.4   Lower Leg:          53.7   Sternal Notch:     138.1   Total Body Height: 171       All the measurements are in cm.   The ranges that are given in the results are these:    Seat tube range c-c:   52.1 - 52.6 for competitive fit.   Seat tube range c-c:   53.3 - 53.8 for eddy fit.    The purpose of my bike is going to be able to do long distance mainly. I cannot afford another bike for racing but I will most probably be doing Ironman from next year using the same bike. But I personally think that riding 180 km in 8 hr is endurance and not year.  Now which size should I go for?  52 or 54? Should I just check the standover height of the bike? Or I should I look into something more than that?  I am asking this because in India not many shops deal with Trek road bikes. The Trek 1.2 is only available at one shop. That fellow says he won't allow me to test ride the bike. So I am in a bad situation. I want to buy this bike but the right size. The whole geometry is available in the site I have given. Ask for more information, if needed.
89414
I'm reading "How would you move Mount Fuji?", and one of the puzzzles/questions is:     A train leaves Los Angeles for New York at a constant   speed of 15 miles an hour. At the same moment, a train   leaves New York for Los Angeles on the same track. It travels   at a constant 20 miles an hour. At still the same moment,   a bird leaves the Los Angeles train station and flies toward   New York, following the track, at a speed of 25 miles an   hour. When it reaches the train from New York, it instantly   reverses direction. It travels at the same speed until it   reaches the train from Los Angeles, when it reverses   again, and so forth. The bird flies back and forth between   the two trains until the very moment they collide. How   far will the bird have travelled?   In the answer, they mention that it could be solved by using an infinite series (and that most people will probably have forgotten how to do it when asked in an interview, and that John von Neumann did so almost instantly when asked this type of question), but they do it another way. First, they note that the time until the trains crash is given by h, and d = 15h + 20h, where d is the distance (they use 3500). Thus, h = 100 hours. The distance that the bird travels is 25d. Thus, the answer is 2500 miles.  I'm curious though, how does one solve this using an (infinite) series? How does one write this problem as a series in the first place? The only thing that I can get is that the series (of the hours that the bird spends flying) starts like this: (77.77, 9.73, ...)
66115
A monk's Unarmored Movement class feature increases the monks speed by 10 feet at level 2, and further at higher levels. According to , this stacks with an Aarakocra monk's fly speed, because that fly speed is innate to the race. However, how does this interact with fly speeds gained from other sources, such as the Fly spell? That spell specifically states that &quot;the target gains a flying speed of 60 feet for the duration.&quot; Can this gained flying speed then be improved by a monk's Unarmored Movement, giving a level 2 monk a flying speed of 70 feet while the Fly spell is active on them?
28261
I have heard that one ought to turn the pedals at 90rpm. However, I do so much slower, especially while standing, and would like to understand the argument behind this number (or any other number). So:  What is the optimal cadence of a cyclist. Why? Does it depend on the terrain? Does it depend if the rider is sitting or standing upright? What are some possible consequences of deviating too much from the proposed number?  Possible duplicate of , but I didn't find there the "whys" except for "it will grind your knees." However, I find riding upright with slow cadance no more demanding than walking up a hill with a backpack.
89148
What is the distance a creature or player can travel if they have multiple types of movement?  For example, the  has a 40 ft walking speed, a 40 ft climbing speed, and an 80 ft flying speed. Can it move a total of 160 ft in a round using each of those fully, or is there another way this is calculated?
11655
This question acts as a clarification to  on this site. Here's the scenario:   You have 60' of movement at the start of your turn You move 30' to any direction You grapple someone at this point How far do you have left to move?
57418
I recently read a math problem and, having not yet taken anything beyond calculus 1, was curious about how to solve it correctly.   Problem:  Calculate the probability of two people meeting at the same location between 1 and 2 p.m. Assume both people show and person 1 will wait 15 minutes for person 2.     Doesn't this probability increase as the time frame shrinks?  For example, Isn't there a much smaller chance the pair will meet if person 1 arrives at 1:00 vs arrives at 1:45 (the probability of meeting becomes 1).  Is this somewhat easily answered using basic statistics?  Just curious if my thought process is accurate or way wrong.
50098
I just bought a new MTB and am planning to change my previous MTB into a Road Bike.   How do I do it? What kind of parts do I need to change or add or remove?  I am not looking to for a full conversion into specialised road bike. But like the answers are suggesting, I am looking for a bridge between commuter or road bike. My majors concerns to be met are:   Should be fast enough to compete with road racers over short distance Should be light as a road  Should be as close as possible in as cheap as possible.
37367
I am teaching an elementary student. He has a homework as follows.     There are $16$ students who use either bicycles or tricycles. The total   number of wheels is $38$. Find the number of students using bicycles.   I have $3$ solutions as follows.  Using a single variable.  Let $x$ be the number of students in question. The number of students using tricycles is $16-x$. The total number of wheels is the sum of the total number of bicycles times $2$ and the total number of tricycles times $3$.  $$ 2\times x + 3 \times (16-x) = 38 $$  The solution is $x=10$.  Using $2$ variables.  Let $x$ and $y$ be the number of students using bicycles and tricycles, respectively. It implies that  \begin{align} x+y&amp;=16\\ 2x+3y&amp;=38 \end{align}  The solution is $x=10$ and $y=6$.  Using multiples  The multiples of $2$ are $2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22,24, 26,28,30,32,\dotsc$  The multiples of $3$ are $3,6,9,12,15,18,21,24,27,30,33,36,\dotsc$   The possible wheel combinations with format $(\#\text{bicycle wheels}, \#\text{tricycle wheels})$:  $\hspace{6cm} (32,6)$ but there are $18$ students  $\hspace{6cm} (26,12)$  but there are $17$ students  $\hspace{6cm} (20,18)$ there are $16$ students  $\hspace{6cm} (14, 24)$ there are $15$ students  $\hspace{6cm} (8, 30)$ there are $14$ students  $\hspace{6cm} (2,36)$ there are $13$ students  Thus the correct combination is $10$ bicycles and $6$ tricycles.  My question  Is there any other simpler method?
122598
So, I think it'd be cool - for goal setting for example - if I could keep track of my elevation change.  The problem is, the local cyclocomputers I looked at seem to use barometric pressure to determine the change.  This is fine, unless some weather is moving in, which happens not infrequently in the Midwestern United States.  Or do some of them try to use an inclinometer, your speed, and some math you learn in Calculus II to keep track of your change?  That would seem more accurate!  I could try to use a GPS meter, but those things are expensive -  and based on back when I had service to my smart phone, not guaranteed to match my car's odometer for distance even - I've read from some other riders that the distance varies at least slightly from local measurement on a bike as well.   Alternatively, let's say that I ride along a known route.  Is there any good way of plotting that route for elevation change?  Thanks!
42579
Motivation: I'm a casual cyclist (I ride around 4 miles per day, to school and work and back) but I wanted to get into unicycling to casually cycle through parks and the like; nothing too extreme, but just to be able to take it on a bus and cycle around the town.  I was unsure of what to look for in a beginner's unicycle, though.  Main Question: What types of things should a beginner unicyclist look for in their first unicycle?  Is it worthless to buy a unicycle labeled as a "beginner's unicycle"?  I browsed the other unicycle thread, but it didn't explicitly apply to people just starting to unicycle.
121575
I'm looking to buy a single gear bicycle. My experience with bikes is very limited, I haven't owned a bike since I was a kid. I have been mountain biking in the Lake District a few times in the last 3 years or so and enjoyed it a lot. At that time, I just hired a mountain bike.  I would like to buy a bike to to commute to and from work (8 miles/13 KM each way - no steep climbs). I'm tempted by the single gear bicycles because they require low maintenance and fitness wise they can be more of a workout.  My criteria so far is:   Has to have a free wheel,  Chain tensioning mechanism, Brakes on both wheels   Is there anything else a newbie like me should consider before buying a single gear bike? I'm 6'2" and weigh 10 stone.  Appreciate the help.