Solutions for Problem Set 3

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.

Problem 28 (Chapter 2)

Diagram

[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.

Problem 57 (Chapter 2)

[Copy of a solution done using MATHCAD]

Diagram

Quantities Given and Needed

Problem 58 (Chapter 2)

[Diagram only]

Diagram

Quantities Given and Needed

Problem 38 (Chapter 4)

Diagram

Quantities Given and Needed

Problem 53 (Chapter 4)

Diagram

Quantities Given and Needed

Problem 54 (Chapter 4)

Diagram

Quantities Given and Needed

Problem 56 (Chapter 4)

[Copy of a solution done using MATHCAD]

Diagram

Quantities Given and Needed

Problem IV

Diagram

Quantities Given and Needed

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

v_tot = v_r + v_b

y = y₀ + v_{tot, y} t

w = x₀ + v_{tot, x} t

and if the origin is placed where he started rowing, x₀ = 0 and y₀ = 0 .

Looking at the diagram, we see that

v_{r, x} = 0 and v_{r, y} = v_r

v_{b, x} = v_b cos(theta) and v_{b, y} = v_b sin(theta)

If we use these results and substitute for v_tot in terms of v_r and v_b , our working equations are

w = v_b [cos (theta)] T

y = [v_r + v_b sin (theta)] T

The first of these gives

T = w / [v_b cos (theta)]       (b) answer

and the value of T may be substituted into the second equation to give

y = w [v_r + v_b sin (theta) ] / [v_b cos (theta) ]       (a) answer