A river flows due east at 1.50 m/s. A boat crosses the river from the south to t

Question

A river flows due east at 1.50 m/s. A boat crosses the river from the south to the north shore by maintain a constant velocity of 10.0 m/s due north relative to the water. (a) - What is the velocity of the boat relative to the shore? (b) If the river is 300 m wide, how far downstream has the boat moved by the time it reaches the north shore? I created a triangle with 10 on the vertical line and 1.5 on the horizontal line. The hypotenuse is 10.11 m/s. I also knew it would take 30 seconds to cross the stream, and based on the 30 seconds * 1.5m/s determined that it was 45 m to the east.

Explanation / Answer

Define: +x: east, +y: north

Vr = 1.5 m/s

Vb = 10 m/s

w = 300 m Vectors:

v_r = Vr*<1,0>

v_b/r = Vb*<0,1>

Add vectors:

v_b/s = v_b/r + v_r

Substitute

v_b/s = Vb*<0,1> + Vr*<1,0>

Simplify v_b/s =<Vr, Vb>

(A) velocity of boat wrt shore:

v_b/s = <1.5 m/ s, 10m/s>

or

v_b/s = 10.11 m/s at bearing of 8.53 degrees

(B) distance of motion:<x,y>= v_b/s*t

<w, d> = <v_b/s_x*t, v_b/s_y*t>

recall

<w, d> = <Vr*t, Vb*t>

t = w/Vr

substitute

<w, d> = <Vr*(w/Vr), Vb*(w/Vr)>

Simplify

<w,d> = <w, Vb*w/Vr>

Distance downstream:

d = Vb*w/Vr d

= 45 meters (RESULT)

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