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**Unformatted text preview: **article’s position, as in cases where the gravitational acceleration varies with
height. Or the force may vary with velocity, as in cases of resistive forces caused by
motion through a liquid or gas.
Another complication arises because the expressions relating acceleration, velocity, position, and time are differential equations rather than algebraic ones. Differential equations are usually solved using integral calculus and other special
techniques that introductory students may not have mastered.
When such situations arise, scientists often use a procedure called numerical
modeling to study motion. The simplest numerical model is called the Euler
method, after the Swiss mathematician Leonhard Euler (1707 – 1783). The Euler Method
In the Euler method for solving differential equations, derivatives are approximated as ratios of ﬁnite differences. Considering a small increment of time t, we
can approximate the relationship between a particle’s speed and the magnitude of
its acceleration as
a(t) v(t v
t t)
t v(t) t) of the particle at the end of the time interval t is apThen the speed v(t
proximately equal to the speed v (t ) at the beginning of the time interval plus the
magnitude of the acceleration during the interval multiplied by t :
v(t t) v(t) a(t) t (6.10) Because the acceleration is a function of time, this estimate of v(t
t) is accurate
only if the time interval t is short enough that the change in acceleration during
it is very small (as is discussed later). Of course, Equation 6.10 is exact if the acceleration is constant. 6.5 Numerical Modeling in Particle Dynamics The position x(t
t) of the particle at the end of the interval
found in the same manner:
v(t) x
t x(t t)
t x(t t) x(t) t can be v(t) t 1
2 x(t)
(6.11) t)2 You may be tempted to add the term a(
to this result to make it look like
the familiar kinematics equation, but this term is not included in the Euler
method because t is assumed to be so small that t 2 is nearly zero.
If the acceleration at any instant t is known, the particle’s velocity and position
at a time t
t can be calculated from Equations 6.10 and 6.11. The calculation
then proceeds in a series of ﬁnite steps to determine the velocity and position at
any later time. The acceleration is determined from the net force acting on the
particle, and this force may depend on position, velocity, or time:
a(x, v, t) F(x, v, t)
m (6.12) It is convenient to set up the numerical solution to this kind of problem by
numbering the steps and entering the calculations in a table, a procedure that is illustrated in Table 6.3.
The equations in the table can be entered into a spreadsheet and the calculations performed row by row to determine the velocity, position, and acceleration
as functions of time. The calculations can also be carried out by using a program
written in either BASIC, C
, or FORTRAN or by using commercially available
mathematics packages for personal computers. Many small increments can be
taken, and accura...

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