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Finding distance traveled...? Mechanics questions

  1. Dec 12, 2015 #1
    1. The problem statement, all variables and given/known data
    A point mass,m, is constrained to move in one-dimension and is acted on buy a force that depends on time in the following way:
    F[x](t)=ƒ[o]+αt+βt^2
    where ƒo,α, and β are constants . In terms of the quantities given, answer the following:
    If the object starts off at rest at t=0, find its velocity at a later time, t[f]
    Find the distance the mass has traveled from t=0 to t=t[f]
    2. Relevant equations
    v=at
    F=ma

    3. The attempt at a solution
    I believe I found the velocity at t[f] by, having
    a=F/m then
    v=(F/m)t
    and for t[f]
    v[f]=((ƒ[o]+αt[f]+βt[f]^2)/(m))*t[f]


    but from there, I'm kinda stuck.
    I know that the distance must be the integral of the velocity, but, with velocity changing, how can I find the integral?
    Or is there perhaps another way?
     
  2. jcsd
  3. Dec 12, 2015 #2

    SteamKing

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    You're partially correct.
    v = at only when a = constant. Is a = constant here?
     
  4. Dec 12, 2015 #3
    a is not a constant, meaning my answer of v[f] is wrong. I could find dv/dt for t[f], but that would only give me the acceleration at that time. How could I find, then, the final velocity at t[f]?
     
  5. Dec 12, 2015 #4

    SteamKing

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    Since a = dv / dt, you'll have to integrate acceleration to find velocity.
     
  6. Dec 12, 2015 #5
    ok. so,
    Fx(0)=f[o]
    Fx=ma
    a=f[o]/m (let this be the first acceleration, or a[1])
    and for t=t[f]
    Fx(tf)=f[o]+αt[f]+βt[f]^2
    Fx=ma
    a=(f[o]+αt[f]+βt[f]^2)/m (let this be acceleration #2, or a[2])
    Δa=a[2]-a[1]
    Δa=(1/m)(αt[f]+βt[f]^2)
    So, I integrate Δa with respect to t?
    but, the boundaries for the integral would be (0.t[f])?
    I'm confused with how to integrate this, or how to evaluate t[f] (either as a value or variable)?
     
  7. Dec 12, 2015 #6

    SteamKing

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    Have you studied calculus any? Do you know how to integrate a function of a single variable?
     
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