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Question about Geometry vs. Calculus in Physics Problem

  1. Sep 25, 2011 #1
    Okay, I'm not asking how to find the solution to this problem. I already found the solution they were looking for. The thing that confuses me is that I got two different solutions using two methods that should have given me the same answer. Could someone show me what I did wrong?

    1. The problem statement, all variables and given/known data

    Find the distance d0,2 traveled by the car between t = 0 s and t = 2 s.

    This graph is given:
    PhysicsQuestion1.jpg

    2. Relevant equations

    Equations:
    For t = 0s to t = 1s: V(t) = 30t
    For t = 1s to t = 2s: V(t) = 20t

    3. The attempt at a solution

    The distance is the area under the curve. You should be able to solve this either with calculus or with geometry.

    I used calculus first. I took the definite integral of both equations and added them together.

    The definite integral from 0 to 1 of 30t is 15. (15(1)^2 - 15(0)^2 = 15)
    The definite integral from 1 to 2 of 20t is 30. (10(2)^2 - 10(1)^2 = 30)
    Adding them together, the total distance traveled should have been 45 m.

    This answer was incorrect. So I tried using geometry.

    The first interval was a triangle. The area of a triangle is (1/2)b*h.
    For 0 to 1: b = 1, h = 30. (1/2)(1*30) = 15. Same as the definite integral.

    The second interval was a triangle on top of a rectangle. The area of a rectangle is b*h.
    For 1 to 2: b = 1, hrectangle = 30, htriangle = 20.
    (1*30) + (1/2)(1*20) = 40. Not the same as the definite integral.

    Adding them together, I got 55, which was the correct answer.

    So, now for my question: Why aren't they the same? Did I make a mistake somewhere, or is my understanding incorrect?
     
  2. jcsd
  3. Sep 25, 2011 #2

    Doc Al

    User Avatar

    Staff: Mentor

    OK, this one matches the diagram.
    But this one doesn't match up. Check a couple of values, such as t = 1 and t = 2.

    Fix that second equation.
     
  4. Sep 25, 2011 #3
    XD Ahhh... so simple. Okay, the equation for the interval is 20t + 10. Thanks.
     
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