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What is the speed?

  1. Jan 29, 2016 #1
    1. The problem statement, all variables and given/known data

    You are in your car driving on a highway at 25 m/s when you glance in the passenger-side mirror (a convex mirror with radius of curvature 150 cm) and notice a truck approaching. If the image of the truck is approaching the vertex of the mirror at a speed of when the truck is 2.0 m from the mirror, what is the speed of the truck relative to the highway?

    2. Relevant equations



    3. The attempt at a solution

    (sorry my bad english)
    Assuming both the truck and the car speeds to be constant, the truck takes the same time interval as its image to reach the vertex of the mirror. So,

    1 / f = 1 / s + 1 / s'
    -1 / 0.75 m = 1 / 2 m + 1 / s'
    s' = -0.54 m
    t = -0.54 m / -1.9 m /s
    [ (V - 25) m / s ] t = 2 m
    but solving this for V I didnt find the correct value: 51 m / s. Maybe the speed is'nt constant?

    My sketch:

    1264etw.jpg
     
  2. jcsd
  3. Jan 29, 2016 #2

    haruspex

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    The speed of the image will not be constant. You can see this by differentiating the 1/f formula.
     
  4. Jan 29, 2016 #3
    I dont know how to do this :frown:
     
  5. Jan 29, 2016 #4

    haruspex

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    The derivative of s with respect to time, ds/dt, is the velocity, v. What is the derivative of 1/s? Use the chain rule.
     
  6. Jan 29, 2016 #5
    hmmmm let me try...

    1 / f = 1 / V dt + 1 / V' dt
    1 / f = 1 / dt (1 / V + 1 / V')

    but how can I solve this derivative? would I need to express f as function of t?
     
    Last edited: Jan 29, 2016
  7. Jan 29, 2016 #6

    haruspex

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    f is a constant. What is the derivative of a constant?
    The chain rule says d/dt(1/s)= (ds/dt)(d/ds(1/s)). What is d/ds(1/s)?
     
  8. Jan 31, 2016 #7
    ohh yah

    (d / dt) 1 / f = 0
    0 = - [(V - 25 m / s) / s²] - V' / s'²
    V ≅ 51 m / s

    Thanks :smile:

    but why the image speed is not constant?
     
    Last edited: Jan 31, 2016
  9. Jan 31, 2016 #8

    haruspex

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    Because the relationship between the two distances is nonlinear.
     
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