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Kinematics

  1. Position is specified by coordinates;
    displacement is specified by differences of coordinates;
    distance is specified by the Pythagorean Theorem (and is therefore always positive).

  2. Velocity is the rate of change of position (measured in m/s);
    acceleration is the rate of change of velocity (measured in m/s2).

  3. Δf/Δt is the finite change in f during the finite time interval Δt;
    df/dt (the derivative of f with respect to t) is the infinitesimal change in f at the instant of time t.

  4. Using a Taylor series, any smooth function can be represented in the neighborhood of a point x0 by
    f(x) = Σk=0∞ f (k) (x0) * (x - x0)k / k!,
    where f (k) denotes the kth derivative of f. So
    x(t) = x(0) + dx/dt (0) * t + d2x/d2t (0) * t2 / 2 + d3x/d3t (0) * t3 / 6 + ...
    If d2x/d2t does not depend on t (so dk > 2x/dkt = 0), we write
    x(t) = x(0) + v(0) * t + a * t2 / 2,
    which is the equation for uniformly accelerated motion.

    In general, in order to truncate the Taylor series at any given order, we must establish that all higher order terms are negligible.

  5. For uniformly accelerated motion
    dx(t)/dt = v(t) = v(0) + a * t
    and
    d2x(t)/dt2 = dv(t)/dt = a.

  6. For uniformly accelerated motion
    ∫ a dt = a * t + c1
    = v(t) with c1 = v(0),
    and
    ∫ v(t) dt = v(0) * t + a * t2 / 2 + c2
    = x(t) with c2 = x(0).

  7. Graphically, v(t) is the slope of the graph of x(t) at a specific time, and a(t) is the slope of the graph of v(t) at a specific time.

    Similarly, Δv is the area under the a(t) curve between two times, and Δx is the area under the v(t) curve between two times.

    Hence Δx/Δt is the average velocity, and Δv/Δt is the average acceleration, between two times.

    For constant acceleration, the average velocity is equal to the instantaneous velocity at the mid-point of the time interval over which the average is taken.

    Also, note that total distance travelled is not necessarily the same as total displacement! A graph of v vs. t should clarify the matter.

  8. So with x(t) = x(0) + v(0) * t + a * t2 / 2 and its first time derivative, we have two equations and six unknowns: x(t), x(0), v(t), v(0), a and t.

    Given any four, we can always find the other two; the method of solution is determined by which four variables are given.

    Since there are 15 ways to choose 4 things from a group of 6 (6! / (4! 2!)), there are 15 possible one-dimensional kinematic problems.

  9. Physical considerations useful when analyzing uniform linear motion:

    1. Choose the origin of the coordinate system (frame of reference) to simplify initial positions whenever reasonable. This often means choosing the origin so that x(0) = 0.

    2. Ignoring "internal" motions, the motion of an extended object is equivalent to the motion of its center of mass.

    3. Except when velocities approach the speed of light, or in large gravitational fields, we can use the same time variable for all objects.

    4. If two objects with positions x1(t) and x2(t) collide, x1(tcollision) = x2(tcollision).
      Note that for a collision to occur, the solution(s) to this equation must be real.
      For two objects, we can usually set one of the initial positions to zero by defining it to be the origin. There are therefore five equations and 10 unknowns, yielding 252 possible collision problems (10! / (5! 5!)).

    5. When an object reverses direction, v(treversal) = 0. This happens when what went up starts to come down.

    6. Under the influence of gravity near the surface of the Earth, a = - g = - 9.8 m/s2.

    7. If the acceleration changes from one constant value to another in the same problem, break the problem into two pieces and solve them in turn, using the results of the first part to solve the second part.

  10. The general flow of problem solving in physics is to

    1. Determine what type of problem you are dealing with (ie., projectile, circular motion, etc.).
    2. Draw a careful diagram showing all numerical aspects of the problem.
    3. Make a complete list of all given numerical information, and all unknowns. Assign appropriate variables to all.
    4. Write down the general equations describing the problem.
    5. There should be n equations in n unknowns; solve that system of equations for the unknowns.
    6. Substitute the given (or assumed) values into the solutions.

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©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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