Question

A 400.0-m-wide river flows from west to east at 30.0 m/min. Your boat moves at 100.0 m/min relative to the water no matter which direction you point it. To cross this river, you start from a dock at point A on the south bank. There is a boat landing directly opposite at point B on the north bank, and also one at point C , 75.0 m downstream from B (the figure (Figure 1)).

(a) Where on the north shore will you land if you point your boat perpendicular to the water current?

(b) What distance will you have traveled, if you point your boat perpendicular to the water current?

(c) If you initially aim your boat directly toward point C and do not change that bearing relative to the shore, where on the north shore will you land?

(d) To reach point C at what bearing must you aim your boat?

(e) How long will it take to cross the river?

(f) What distance do you travel?

(g) What is the speed of your boat as measured by an observer standing on the river bank?

Answer 1

Where on the north shore will you land if you point your boat perpendicular to the water current?

Vb = 100/60 = 1.667 m/sec boat speed

t = 400/Vb = 400/1.667 = 240 sec crossing time

Vs = 30/60 = 0.5 m/sec stream speed

drift = Vs*t = 240 /2 = 120 m

What distance will you have traveled?

V = sqrt(Vs^2+Vb^2) = sqrt(0.25+1.667^2) = 1.74 m/sec

d = V*t = 1.74*240 = 417.69 m

If you initially aim your boat directly toward point C and do not change that bearing relative to the shore, where on the north shore will you land?

heading angle ? = arctan (75/400) = 0.185 rad

cos ? = 0.983

V_I_ = Vb* 0.983 = 1.64 m/sec

t = 400/1.64 = 244.1 sec

drift from C = Vs*t = 244.1 /2 = 122.05 m

drift from B = 122.05+75 = 197.05 m

To reach point C at what bearing must you aim your boat?

Vy*t = 400 ..(1)

Vs*t-Vx*t = 75

(0.5-Vx)*t = 75...(2)

Vx^2+Vy^2 = 3.361..(3)

(3.361-Vx^2)*t^2 = 400^2

t = 75/(0.5-Vx)

(3.361-Vx^2)*75^2/(0.25^2+Vx^2-Vx) = 400^2

3.361-Vx^2 = 7,111+28.44Vx^2-28.44Vx

29.44Vx^2-28.44Vx+3.750 = 0

Vx = (28.44-?28.44^2-29.44*15)/58.88 = 0.1576 m/sec

Vy = sqrt(3.361-0.0248) = 1.8265 m/sec

t = 400/Vy = 400/1.8265 = 219.0 sec

bearing = arctan Vx/Vy = arctan 0.1576/1.8265 = 4.93°

check : C drift referred to B = (Vs-Vx)*t = (0.5-0.1576)*219 = 75.0 m!!!

How long will it take to cross the river?

t = 400/Vy = 400/1.8265 = 219.0 sec

What distance do you travel?

d = sqrt(400^2+75^2) = 407.0 m

What is the speed of your boat as measured by an observer standing on the river bank?

V = 407/219 = 1.858 m/sec