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    time question

    If it is currently 5:15. At what time down to the exact fraction of a second will the minute hand and hour hand meet? thanks!
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    Quote Originally Posted by curious444 View Post
    If it is currently 5:15. At what time down to the exact fraction of a second will the minute hand and hour hand meet? thanks!
    every 5 minute increment on the face of an analog clock is \frac{\pi}{6} radians.

    let the 3 o'clock position be the zero position for both hands.

    at t = 0 (5:15) the minute hand is at \theta = 0 and is moving with speed \omega = 2\pi radians per hr.

    at t = 0, the hour hand is at \phi = \frac{\pi}{3} + \frac{\pi}{24} = \frac{3\pi}{8} radians, moving at \frac{\pi}{6} radians per hr.

    for t in hours, the two hands coincide when their respective positions are the same, \theta = \phi ...

    0 + 2\pi \cdot t = \frac{3\pi}{8} + \frac{\pi}{6} \cdot t

    2t = \frac{3}{8} + \frac{t}{6}

    \frac{11t}{6} = \frac{3}{8}

    t = \frac{18}{88} = \frac{9}{44} hrs

    \frac{9}{44} hrs = 12 min + 16.363636... sec

    the hands coincide at 5:27 + 16 and \frac{12}{33} sec
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