This problem considers a way to compute velocity and position that differs from the one considered in Exercise 6.15. You will find that the value of
in part (c) is a factor of about 0.07 smaller than the corresponding value from Exercise 6.15(c).
(a) Suppose the interval 0 ≤
≤ 3 is subdivided into
equally spaced subintervals. So,
i
= (
− 1)
, where
= 1, 2, 3, ··· ,
+ 1 and
= 3/.
Explain how the Hermite rule can be used to obtain the following expressions
(b) Assuming that
() = sin(
4), plot, on the same axis,
as a function of
, for
= 10, 20, 40.
(c) An accurately computed value for the position at
= 3 is
(3) = 0.72732289075 ..... What is the difference between this value and what you compute for
(3) at
= 10, 20, 40? How large does
need to be so that this value and what you compute for
(3) is less than 10−8
in absolute value?
Exercise 6.15
The position
(), velocity
(), and acceleration
() are related through the equations:
() =
’() and
() =
’(). In this problem it is assumed that
(0) = 0 and(0) = 0. In this case,
It is also assumed that
() is known, and the objective of this exercise is to compute the velocity and position from this information.
(a) Given a subinterval
i
≤
≤
i+1, then
i
=
(
i
) and
i+1
=
(
i+1) are known. Assuming
i
and
i
have already been computed, use the trapezoidal rule to obtain the following expressions
(b) Suppose the interval 0 ≤
≤ 3 is subdivided into
equally spaced subintervals. So,
i
= ( − 1), where
= 1, 2, 3, ··· ,
+ 1 and
= 3/. Assuming that
() = sin(
4), plot
as a function of
, for
= 10, 20, 40. The three curves should be on the same axis.
(c) An accurately computed value for the position at
= 3 is
(3) = 0.72732289075 ..... What is the difference between this value and what you compute for
(3) at
= 10, 20, 40? How large does
need to be so that this value and what you compute for
(3) is less than 10−8
in absolute value?