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Acceleration and distance!

  1. Aug 31, 2012 #1
    Hi there everyone! I'm a first-year physics student and I need some help with a couple of questions! OK, here it goes :)

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

    Task 1​
    The acceleration of a vehicle is given by a(t) = αT, where α=1.4 m/s^3. At the time t=1.0 s
    the car has a speed of 5.0 m/s, what is the velocity at t=3.0 s?

    (α = alpha) (In other words a(t)=α=1.4 m/s^3 is not a typo)

    There are four possible answers:
    (a) y (3) = 6.3 m / s
    (b) v (3) = 7.4 m / s
    (c) y (3) = 8.7 m / s
    (d) y (3) = 11 m / s

    Task 2​
    We use with the same car as in the previous problem. We learn that the car
    position at t = 1,0s is 5.0 m Find the car's position at t = 3.0 s

    There are four possible answers:
    (a) x (3) = 15m
    (b) x (3) = 18m
    (c) x (3) = 20m
    (d) x (3) = 23m

    2. Relevant equations

    The relevant equations would be the kinematic equations for constant acceleration:
    V = Vo + at
    X = Xo + VoT+ 0.5aT^2
    V^2=Vo^2 + 2a(-Xo)

    3. The attempt at a solution

    Task 1​
    V = Vo + at

    V = 5 m/s + 1,4 m/s^2 *2 s (the difference between the two t-values is 2)
    V = 7,8 m/s

    This answer is not one of the possible solutions --> I have done something wrong
    Possible solutions (again):
    (a) y (3) = 6.3 m / s
    (b) v (3) = 7.4 m / s
    (c) y (3) = 8.7 m / s
    (d) y (3) = 11 m / s

    Task 2​

    X=VoT + 0,5at^2

    X=5 m/s x 2 s + 0,5*1,4 m/s^2 * (2s)^2 (Again, the difference between the t-values = 2)
    X = 12,8 m

    The task specifies that the distance at T(1)=5 --> Total distance = 5 m + 12,8 m = 17,8

    Again, ther are four possible answers, and I'm fairly sure that the answer is b:
    (a) x (3) = 15m
    (b) x (3) = 18m
    (c) x (3) = 20m
    (d) x (3) = 23m
    Nevertheless, I am a bit unsure about my calculation, and posted this Task as well :)

    Thanks a lot for your help! :D
     
  2. jcsd
  3. Aug 31, 2012 #2

    ehild

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    Hi Warcus. Welcome to PF. :smile:

    In the formula for the acceleration you meant t instead of T I think.

    So a(t)=αt, and α=1.4 m/s3, so a(t)=1.4t.
    This is the case of non-uniform acceleration, your relevant equations are irrelevant :tongue2:.

    Go back to the definition of acceleration: it is the time derivative of velocity. a=dv/dt. The opposite is also true: the velocity is obtained by integrating the acceleration: v(t)= ∫a(t)dt, and adding an arbitrary constant. The constant can be found from the condition that v=5.0 m/s at t=1.0 s.

    ehild
     
  4. Sep 1, 2012 #3
    Hi and thanks for the assistance ehild! I tried doing what you said, but as this is my first year studying physics I encountered some problems... Take a look, will you?

    v(t) = ∫a(t)dt
    v(t) = (t^a) * t + C (where a is 1,4t)
    v(3) = (3^1,4*3) * 3 + C = 9,12 + C

    Vo = V(1) = 1,4(1) + C --> C = 3,6

    v(3) = 12,72

    which can't be right according to the answers the teacher gave me.. :confused:

    Could you try working it out step-by-step so I can see how it is done? :rolleyes:
     
  5. Sep 1, 2012 #4

    ehild

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    The acceleration is a=1.4 t. v(t) is the integral or primitive function of the acceleration. V(t)=∫(1.4 t dt) = 1.4 ∫(t dt) =1.4 t2/2+C. You have studied some calculus , about integration, have you not?
    Now substitute t=1 and v=5 and you get C.
    Use that value of C to get the velocity at t=3 s.


    ehild
     
  6. Sep 1, 2012 #5
    V(t) = (1.4*t2)/2 + C
    C = V(t) - (1.4*t2)/2
    C = V(1) - (1.4*12)/2
    C = 5 - (1,4 *12)/2
    C = 4,3


    V(t) = (1.4*t2)/2 + C
    V(3) = (1.4*32)/2 + 4,3
    V(3) = 10,6
    V(3) ≈ 11
    V(3) = answer number d

    And yeah, I studied some integration, but hey... It was three years ago and was probably just the basics. Anyways, thanks for the help, it was greatly appreciated! :biggrin:
     
  7. Sep 1, 2012 #6

    ehild

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    Do not stop yet. Task 2 says that it is the same car as before, which has the velocity (you know it already) V(t) = (0.7*t2 + 4.3. You need the position as function of time. It is the integral (antiderivative) of the velocity.

    [tex]x=\int{v(t)dt}=\int{(0.7t^2+4.3)dt}[/tex]

    Refresh your knowledge about the integrals (antiderivatives) If it is too difficult, you can cheat a bit.

    http://calculator.tutorvista.com/math/585/antiderivative-calculator.html#
    When you know the expression for x(t), you can find the unknown C again from the condition x=5.0 m at t=1.0 s.

    ehild
     
  8. Sep 1, 2012 #7
    Alright, task 2:

    x = ∫v(t)dt = ∫(0.7t2+4.3)dt = (0.7t3)/3 + 4.3t + C

    x = (0.7t3)/3 + 4.3t
    x(1) = (0.7*13)/3 + 4.3*1 + C
    5 = (0.7*13)/3 + 4.3*1 + C
    C = 5 - 4.53333... = 0.4666...

    x = (0.7*33)/3 + 4.3*3 + C
    x = (0.7*33)/3 + 4.3*3 + 0.46666...
    x = 19.6333
    x ≈20 → Answer C

    Thanks for helping me get the answers and teaching me to antideriviative!! :biggrin: :biggrin:
     
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