Last revised 1999/09/24 |
HRW pp. 28-35 #24,
28,
57,
58 diagram,
72
HRW pp. 72-80 #
38 diagram,
53 diagram,
54,
56
Sheet #
IV
Files referenced
A number of the solutions, both for this week and for future sets, were first generated using MATHCAD, which includes parts of the MAPLE symbolic-algebra package. You should learn to use one of the packages as soon as possible, preferably the one used in your math classes. I find version 8 of MATHCAD very powerful but somewhat unintuitive and difficult to use. If you choose to learn MATHCAD, you might prefer to use an earlier version.
[The diagrams were generated with MATHCAD.]
Since x is given as x(t) = 20 t - 5 t3 , v and a can be obtained by differentiation. The results,
v(t) = 20 - 15 t2
a(t) = - 30 t
are plotted above, and the numerical results requested are readily obtained.
(a) v = 0 implies 20 - 15 t2 = 0 so that t = (2/3) sqrt(3) or t = - (2/3) sqrt(3)
(b) a=0 implies - 30 t = 0, so t = 0
(c) From the graph, a(t)<0 when t>0, and a(t)>0 when t<0.
(d) The graphs are shown above.
Velocity of water in river | vr , known |
Magnitude of velocity of boat relative to water | vb , known |
Rowing angle | theta, known |
Velocity of boat relative to bank | vtot |
Time of travel | T |
Total distance travelled | d |
Width of river | w = dx , known |
Distance down the river | y = dy , requested |
There are two physical situations involved: a relative velocity problem to get the speed of the rower relative to the bank, and unaccelerated one-dimensional motion in the direction of the river and the direction perpendicular to the river. Taking y along the river, positive downward, and taking x perpendicular to the river, we have
vtot = vr + vb
y = y0 + vtot, y t
w = x0 + vtot, x t
and if the origin is placed where he started rowing, x0 = 0 and y0 = 0 .
Looking at the diagram, we see that
vr, x = 0 and vr, y = vr
vb, x = vb cos(theta) and vb, y = vb sin(theta)
If we use these results and substitute for vtot in terms of vr and vb , our working equations are
w = vb [cos (theta)] T
y = [vr + vb sin (theta)] T
The first of these gives
T = w / [vb cos (theta)] (b) answer
and the value of T may be substituted into the second equation to give
y = w [vr + vb sin (theta) ] / [vb cos (theta) ] (a) answer
Image files referenced: 2-28a 2-57a, 2-57b, 2-57c, 2-57d, 2-57e, 2-57f, 2-57g 2-58a 2-72b, 2-72c, 2-72d, 2-72e 4-38a 4-53a 4-54a 4-54b, 4-54c, 4-54d, 4-54e 4-56a, 4-56b, 4-56c, 4-56d, 4-56e, 4-56f, 4-56g IV |