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Relative-Motion analysis between two particles

  1. Jun 6, 2006 #1
    At the instant shown, the bicyclist shown at A is traveling at 7 m/s around the curve while increasing his speed at 0.5 m/s^2. The bicyclist at B is traveling at 8.5 m/s along the straight-a-way and increasing his speed at 0.7 m/s^2. Determine the relative vrlocity and the relative acceleration of A with respect to B at this instant.

    [​IMG]

    I figured out the relative velocity:

    [tex]\overrightarrow{V_B}\,=\,\left{(8.5)\,\widehat{i}\right}\,\frac{m}{s}[/tex]

    [tex]\overrightarrow{V_A}\,=\,\left{(7\,cos\,50)\,\widehat{i}\,+\,(-7\,sin\,50)\,\widehat{j}\right}\,\frac{m}{s}[/tex]

    [tex]\overrightarrow{V_{A/B}}\,=\,\overrightarrow{V_A}\,-\,\overrightarrow{V_B}[/tex]

    [tex]\overrightarrow{V_{A/B}}\,=\,\left{[\,(7\,cos\,50)\,-\,8.5]\,\widehat{i}\,+\,(-7\,sin\,50)\,\widehat{j}\right}\,\frac{m}{s}[/tex]

    [tex]\overrightarrow{V_{A/B}}\,=\,\left{(-4.00)\,\widehat{i}\,+\,(-5.36)\,\widehat{j}\right}\,\frac{m}{s}[/tex]

    [tex]V_{A/B}\,=\,\sqrt{(-4.00)^2\,+\,(-5.36)^2}\,=\,6.69\,\frac{m}{s}[/tex]

    The above answer for the magnitude of the velocity is correct. However, I cannot figure the acceleration part!!!

    [tex](A_A)_N\,=\,\frac{V_A^2}{\rho}\,=\,\frac{7^2}{50}\,=\,0.98\,\frac{m}{s^2}[/tex]

    I know the normal part of the bike at A's acceleration, but when I use this and the other EQ's for the acceleration, I don't get the correct answer! Please Help!

    [tex]a_B\,=\,a_A\,+\,a_{B/A}[/tex]
     

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    Last edited: Jun 6, 2006
  2. jcsd
  3. Jun 6, 2006 #2

    andrevdh

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    The radial part of the acceleration of A, [tex]a_r[/tex] is given by
    [tex]\frac{v^2}{r}[/tex] while the tangential acceleration of A is [tex]a_t\ =\ 0.5\ m/s^2[/tex] which will itself be normal to the indicated radius. This means that you need to decompose both these accelerations into x- and y-components to get the components of the acceleration of A. Much less work need to be done if you rather align your x/y axis with these two acceleration components [tex]a_r , a_t[/tex] and express the acceleration of B in this sytem.
     
    Last edited: Jun 6, 2006
  4. Jun 6, 2006 #3
    Thanks for the help!

    I set up the coordinate system like you said and get the right answer:

    [tex]\overrightarrow{A_{A}}\,=\,{(0.5)\,\widehat{i}\,+\,(-0.98)\,\widehat{j}}\,\frac{m}{s^2}[/tex]

    [tex]\overrightarrow{A_{B}}\,=\,{\left(0.7\,cos\,50\right)\,\widehat{i}\,+\,\left(0.7\,sin\,50\right)\,\widehat{j}}\,\frac{m}{s^2}[/tex]

    [tex]\overrightarrow{A_{A/B}}\,=\,\overrightarrow{A_A}\,-\,\overrightarrow{A_B}[/tex]

    [tex]\overrightarrow{A_{A/B}}\,=\,{(0.05)\,\widehat{i}\,+\,(-1.52)\,\widehat{j}}\,\frac{m}{s^2}[/tex]

    [tex]A_{A/B}\,=\,\sqrt{\left(0.05\,\frac{m}{s^2}\right)^2\,+\,\left(-1.52\,\frac{m}{s^2}\right)^2}\,=\,1.52\,\frac{m}{s^2}[/tex]
     
  5. Jun 6, 2006 #4

    andrevdh

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    Its a privilege to help someone who enjoys his physics and gives me great pleasure.
     
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