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Kinematics with Calculus

  1. Jul 2, 2011 #1
    I just started working through Halliday and Resnick, and I'm going through the kinematic equations for constant acceleration. I don't need help with the work, but I noticed that the book never addresses non-constant acceleration. I was expecting more Calculus in this book (the kinematic equations were derived using algebra), and was wondering where I could find learning material for kinematics with non-constant acceleration. I also wanted to know if I got the right book? I see bits of Calculus throughout, but I'm worried that the majority of the book will deal with special cases and assumptions like the constant acceleration in this chapter. It seems kind of useless to keep looking at special cases like that.
  2. jcsd
  3. Jul 2, 2011 #2
    It all comes from the definitions of position and velocity:

    a = dv/dt and v = dx/dt. It's more useful to write in differential form:

    dv = adt
    dx = vdt

    For constant acceleration you can integrate the first equation to get:

    v = at + const.

    Plugging in t = 0 gives you the constant vo.

    v = vo + at.

    You can then plug this into the dx equation and integrate to get:

    x = vot + 1/2 at^2 + const. (The constant is x0.)

    You follow the same procedure for non-constant acceleration. If a is a function of time you simply integrate. If a is a function of x you can use the chain rule and integrate.

    a(x) = dv/dt = dv/dx dx/dt = dv/dx v

    Writing in differential form,
    a(x) dx = v dv

    or 1/2 v^2 = Int (a(x) dx)

    Assuming you can integrate the function, you can find a relation for v as a function of x. You can then integrate v to find x as a function of time. Another important case is when the acceleration is a function of v.

    a(v) = dv/dt -> dt = dv/a(v)

    If you integrate, you will have time as a function of velocity. You may or may not be able to solve for v as a function of t.

    Surprisingly, it's hard to find an introductory physics book that really uses calculus from the beginning. HRK is pretty good overall. I like their style and they covers a lot of material.

    (Sorry I don't know how to make nice equations.)
  4. Jul 2, 2011 #3
    Thanks! That's exactly the sort of thing I'm looking for, that was great! Where would I go to find more of that kind of thing? You said HRK is good, but if it doesn't use calculus in the beginning, where do I learn a calculus-based approach to these topics? Also, am I to take it that HRK focuses more on calculus later on?
  5. Jul 2, 2011 #4
    Just to be clear, I'm using the 4th ed of HRK, simply titled Physics. Halliday, Resnick, and Walker wrote a newer book called Fundamentals of Physics that I am not familiar with. Chapter 6 of HRK pretty much has what you are looking for. In section 6.4 they show you the "calculus way" to derive the kinematic equations. They deal with time-dependent and velocity dependent acceleration (sections 6.5 and 6.7).

    There is also a note saying that sections 6.4-6.7 require integral calculus, so you may need to skip or postpone reading until you are more familiar with it. The book is written for students that have recently learned calculus, or are just starting the subject.

    So the book does use calculus, it kind of eases you into it though. For example, they explain work by showing you how to break it up into little pieces, adding them up, and then taking the limit. If you already know calculus it's a bit wordy, but they will get you there. Don't worry though. Later on in the book they expect you to know calculus.

    If you are using that newer book, and they don't have what you are looking for, you may want to check out the older one by HRK. You can get a good copy on Amazon for less than $1.
  6. Jul 2, 2011 #5
    Great! Thanks! I'm actually using that exact same edition. I turned to 6.4 and there it was. Sorry about that, I just got worried when I flipped through the first few chapters. I've taken integral calculus, and was worried that the book was glossing over it. I see now that they return to kinematics later in the book. I'll just stick with it then. I guess if I'd just looked at the book more carefully, I wouldn't have needed to post. Sorry about that, but I really appreciate the help.
  7. Jul 3, 2011 #6
    Here you are sir:

    [tex]v(t) ~= ~v_0+\int_{t_0}^{t}a(t)dt[/tex]
    [tex]s(t) ~=~ s_0 + \int_{t_0}^{t} v(t) dt ~= ~s_0 + v_0\Delta t +\int_{t_0}^{t} \int_{t_0}^{t}a(t)dt^2[/tex]

    Those equations pretty much sum up kinematics.
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