Results 1 to 3 of 3

Math Help - problem involving differencial equation

  1. #1
    Member
    Joined
    Oct 2006
    Posts
    84

    problem involving differencial equation

    I found this calc problem in a chapter on improper integration, and it is confusing me so any help would be appreciated!

    A car is travelling at 55 mph. The driver sees a traffic jam ahead and hits the brakes. Brakes apply friction. The car's velocity satisfies the differential equation v' = -1080v miles per hour per hour. How far does the car go after the brake is applied?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,937
    Thanks
    338
    Awards
    1
    Quote Originally Posted by clockingly View Post
    I found this calc problem in a chapter on improper integration, and it is confusing me so any help would be appreciated!

    A car is travelling at 55 mph. The driver sees a traffic jam ahead and hits the brakes. Brakes apply friction. The car's velocity satisfies the differential equation v' = -1080v miles per hour per hour. How far does the car go after the brake is applied?
    v' + 1080v = 0

    This is a homogeneous equation and has a solution:
    v(t) = Ae^{-1080t} + B

    So
    v'(t) = -1080Ae^{-1080t}

    -1080Ae^{-1080t} + 1080Ae^{-1080t} + 1080B = 0

    Thus B = 0 and v(t) = Ae^{-1080t}

    v(0) = 55, thus A = 55.

    v(t) = 55e^{-1080t} (in units of mph.)

    Thus:
    x(t) = Int[v(t'), 0, t] = Int[55e^{-1080t'}, 0, t]

    x(t) = 55/(-1080) * e^{-1080t'} |_{t' = 0}^{t' = t}

    x(t) = -(55/1080)*(e^{-1080t} - 1)
    (Note: t is in hours and x(t) is in miles.)

    Now, what is x(0)?
    x(0) = 0.

    So the distance traveled by the car is simply x(t). Now, when does the car stop? The answer is NEVER!! (Look at v(t).) So we need to evaluate x(t) as t --> (infinity). So the distance traveled by the car is:
    d = lim[x(t), t --> (infinity)] = lim[-(55/1080)*(e^{-1080t} - 1), t --> (infinity)]

    d = -(55/1080)*(0 - 1) = 55/1080 miles

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member ecMathGeek's Avatar
    Joined
    Mar 2007
    Posts
    436
    Quote Originally Posted by topsquark View Post
    v' + 1080v = 0

    This is a homogeneous equation and has a solution:
    v(t) = Ae^{-1080t} + B

    So
    v'(t) = -1080Ae^{-1080t}

    -1080Ae^{-1080t} + 1080Ae^{-1080t} + 1080B = 0

    Thus B = 0 and v(t) = Ae^{-1080t}

    v(0) = 55, thus A = 55.

    v(t) = 55e^{-1080t} (in units of mph.)

    Thus:
    x(t) = Int[v(t'), 0, t] = Int[55e^{-1080t'}, 0, t]

    x(t) = 55/(-1080) * e^{-1080t'} |_{t' = 0}^{t' = t}

    x(t) = -(55/1080)*(e^{-1080t} - 1)
    (Note: t is in hours and x(t) is in miles.)

    Now, what is x(0)?
    x(0) = 0.

    So the distance traveled by the car is simply x(t). Now, when does the car stop? The answer is NEVER!! (Look at v(t).) So we need to evaluate x(t) as t --> (infinity). So the distance traveled by the car is:
    d = lim[x(t), t --> (infinity)] = lim[-(55/1080)*(e^{-1080t} - 1), t --> (infinity)]

    d = -(55/1080)*(0 - 1) = 55/1080 miles

    -Dan
    I'm glad you went through this. I solved it, and got that the car never stops, and so deleted my solution thinking it was wrong. I just didn't think to solve it as t -> infinity.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Differencial equation problem?
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: October 17th 2011, 04:33 PM
  2. differencial equation
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: January 10th 2009, 07:32 AM
  3. differencial equation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 5th 2008, 10:31 AM
  4. calc problem differencial equation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 26th 2007, 06:47 PM
  5. Replies: 0
    Last Post: December 16th 2006, 02:23 PM

Search Tags


/mathhelpforum @mathhelpforum