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(0 to t) v dt = ∫ (s1 to s2) ds
(0 to t) [v 1 / (1 + v 1 k t)] dt = ∫ (s1 to s2) ds
ln(1 + v 1 k t) / k = s 2 - s 1
If s 1 equals 0, then:
s = ln(1 + v 1 k t) / k
Note that in this equation, ln is the natural logarithm operator.
This example demonstrates the relative complexity of nonconstant acceleration prob‐
lems versus constant acceleration problems. It's a fairly simple example where you are
able to derive closed-form equations for velocity and displacement. In practice, however,
there may be several different types of forces acting on a given body in motion, which
could make the expression for induced acceleration quite complicated. This complexity
would render a closed-form solution like the preceding one impossible to obtain unless
you impose some simplifying restrictions on the problem, forcing you to rely on other
solution techniques like numerical integration. We'll talk about this sort of problem in
greater depth in Chapter 11 .
2D Particle Kinematics
When we are considering motion in one dimension—that is, when the motion is re‐
stricted to a straight line—it is easy enough to directly apply the formulas derived earlier
to determine instantaneous velocity, acceleration, and displacement. However, in two
dimensions, with motion possible in any direction on a given plane, you must consider
the kinematic properties of velocity, acceleration, and displacement as vectors.
Using rectangular coordinates in the standard Cartesian coordinate system, you must
account for the x and y components of displacement, velocity, and acceleration. Essen‐
tially, you can treat the x and y components separately and then superimpose these
components to define the corresponding vector quantities.
To help keep track of these x and y components, let i and j be unit vectors in the x- and
y-directions, respectively. Now you can write the kinematic property vectors in terms
of their components as follows:
v = v x i + v y j
a = a x i + a y j
If x is the displacement in the x-direction and y is the displacement in the y-direction,
then the displacement vector is:
s = x i + y j
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