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Math Help - Developing a formula for speed of a bicycle

  1. #1
    Junior Member surffan's Avatar
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    Developing a formula for speed of a bicycle

    A bicycle has two sets of gears, one at the pedals and two at the rear wheel. a typical 10-speed bicycle has two gears on the chain wheel and five gears on the rear wheel. the speed at which a bicycle travels depends on three independent factors :

    • The first is the speed at which the cyclist pedals to furn the front gear (rotations per minute)
    • the second is the gear ration fron the front gear to the rear wheel ( ratio between number of teeth on the front gear compared to the number of teeth on the rear gear.)
    • the third is the size of the rear wheel (measured as the diameter)

    a) develope a formula that predicts the velocity in km/h for a bicyclist.
    b) how fast will a cyclist travel (km/h) who is pedalling at 50 rpm with a 42-toothed gear on the front and a 14 toothed gear on the back, and a tire with a diameter of 26 inches.

    I can't say I even have an idea where to start here, My text doesnt really explain how to solve this type of question so even if i can get a general idea of what steps I should follow. Thank you
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  2. #2
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    Quote Originally Posted by surffan View Post
    A bicycle has two sets of gears, one at the pedals and two at the rear wheel. a typical 10-speed bicycle has two gears on the chain wheel and five gears on the rear wheel. the speed at which a bicycle travels depends on three independent factors :

    • The first is the speed at which the cyclist pedals to furn the front gear (rotations per minute)
    • the second is the gear ration fron the front gear to the rear wheel ( ratio between number of teeth on the front gear compared to the number of teeth on the rear gear.)
    • the third is the size of the rear wheel (measured as the diameter)

    a) develope a formula that predicts the velocity in km/h for a bicyclist.
    b) how fast will a cyclist travel (km/h) who is pedalling at 50 rpm with a 42-toothed gear on the front and a 14 toothed gear on the back, and a tire with a diameter of 26 inches.

    I can't say I even have an idea where to start here, My text doesnt really explain how to solve this type of question so even if i can get a general idea of what steps I should follow. Thank you
    I'll answer b). You can develop the genral equation if you "translate" my calculation into an equation with variables.

    The perimeter of the wheel is p = 26 \cdot \pi\ inches = 2.0747\ m
    Let s denote the speed of the byciclist in \tfrac{km}h.

    s = \dfrac{50 \tfrac1{min} \cdot \frac{42\ teeth}{14\ teeth} \cdot 2.0747\ m}{\tfrac{1\ h}{60\ min} \cdot 1000 \frac{m}{km}} \approx 18.67\ \tfrac{km}h
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  3. #3
    Junior Member BugzLooney's Avatar
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    nice...
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  4. #4
    Junior Member surffan's Avatar
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    Quote Originally Posted by earboth View Post
    I'll answer b). You can develop the genral equation if you "translate" my calculation into an equation with variables.

    The perimeter of the wheel is p = 26 \cdot \pi\ inches = 2.0747\ m
    Let s denote the speed of the byciclist in \tfrac{km}h.

    s = \dfrac{50 \tfrac1{min} \cdot \frac{42\ teeth}{14\ teeth} \cdot 2.0747\ m}{\tfrac{1\ h}{60\ min} \cdot 1000 \frac{m}{km}} \approx 18.67\ \tfrac{km}h
    So i walked away from the computer and gave it another try and was happy to see that i ended up with the same answer you stated, I wasn't sure if I had done it right but it turns out so. thanks again
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    Quote Originally Posted by earboth View Post
    I'll answer b). You can develop the genral equation if you "translate" my calculation into an equation with variables.

    The perimeter of the wheel is p = 26 \cdot \pi\ inches = 2.0747\ m
    Let s denote the speed of the byciclist in \tfrac{km}h.

    s = \dfrac{50 \tfrac1{min} \cdot \frac{42\ teeth}{14\ teeth} \cdot 2.0747\ m}{\tfrac{1\ h}{60\ min} \cdot 1000 \frac{m}{km}} \approx 18.67\ \tfrac{km}h
    How did you get 2.0747 m ? I got 0.65 (26 x 2.5/100).

    After calculating your formula, I also got a very different answer...

    0.00518675, is the answer I got.
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  6. #6
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    Quote Originally Posted by Pupil View Post
    How did you get 2.0747 m ? I got 0.65 (26 x 2.5/100).

    After calculating your formula, I also got a very different answer...

    0.00518675, is the answer I got.
    Let's work it out simply. First work out the circumference in inches, but we'll keep pi in there so we don't have to deal with decimals. If the circumference is:
    p = 26 \cdot \pi
    Then we convert it to meters:
    p = 26\pi\ inches \cdot \frac{0.0254 m}{1\ inch} = 2.0747\ m
    Note how I cancelled the inches, leaving only the unit of meters. Also note that I've included more decimal points than I should. If I assume 26 inches is an exact number, I used 3 significant digits (the 0.0254) in my conversion, so I'd call it 2.07 meters.
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    Quote Originally Posted by Grep View Post
    Let's work it out simply. First work out the circumference in inches, but we'll keep pi in there so we don't have to deal with decimals. If the circumference is:
    p = 26 \cdot \pi
    Then we convert it to meters:
    p = 26\pi\ inches \cdot \frac{0.0254 m}{1\ inch} = 2.0747\ m
    Note how I cancelled the inches, leaving only the unit of meters. Also note that I've included more decimal points than I should. If I assume 26 inches is an exact number, I used 3 significant digits (the 0.0254) in my conversion, so I'd call it 2.07 meters.
    Alright, I follow you so far.

    Now with the formula, I get: 5.18675/1000 which doesn't give 18.67. That is where I'm stuck. Sorry if this may come off as noobish to you, I am just frustrated I got this in grade 12 math when I haven't even taken physics yet.
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  8. #8
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    Quote Originally Posted by Pupil View Post
    Alright, I follow you so far.

    Now with the formula, I get: 5.18675/1000 which doesn't give 18.67. That is where I'm stuck. Sorry if this may come off as noobish to you, I am just frustrated I got this in grade 12 math when I haven't even taken physics yet.
    1. Which model of calculator do you use?

    2. What exactly did you type into the calculator?
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    Quote Originally Posted by earboth View Post
    1. Which model of calculator do you use?

    2. What exactly did you type into the calculator?
    1. I use T1-83 plus edition.

    2. 50/60 x 3 x 2.0747/1000 (I assumed the 1h/60 mins just cancels out). I know this may look noobish, but it really confuses me and thanks for all the help thus far.
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  10. #10
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    Quote Originally Posted by Pupil View Post
    1. I use T1-83 plus edition.

    2. 50/60 x 3 x 2.0747/1000 (I assumed the 1h/60 mins just cancels out). I know this may look noobish, but it really confuses me and thanks for all the help thus far.
    1. With a TI83 you must write divisions (fractions) in brackets.

    2. \underbrace{(50*(42/14)*2.04747)}_{numerator} / \underbrace{(1/60 * 1000)}_{denominator}

    3. By the way \dfrac{n}{\frac1{60}} = 60 \cdot n
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