Someone posted a response to this earlier but I don\'t think the answer was corr

Question

Someone posted a response to this earlier but I don't think the answer was correct... Can someone please take the time to help me carefully with this? Thanks!

A kayaker can paddle her kayak at a steady 2.5m/s in still water. She wishes to cross a river that is 2000m wide and has a current of -1.5m/s (goes from right to left along x axis). She is crossing going "north", along the positive y axis.

(a) IF the kayaker first "aims" her craft straight across the river, the current will carry her downstream as she paddles across. What will be her actual velocity (magnitude and direction angle) as she crosses? How long will it take her to cross the river?

(b) If she "aims" the kayak somewhat upstream, she can actually travel straight accross the river. In what direction must she aim? What is her actual speed accross the river for this situation, and how long will it take her to cross?

HINT: The angle found in (a) will NOT be the same angle required in part (b).

--- I have figured out (a) to be: a velocity of 2.92 m/s at 121 degrees (CCW from x axis). I figured it would take 805s to cross (the distance she travels, or the hypotonuse of the triangle, is 2333.27m). I really need help on part (b) though :(

Is that right? And could someone help me on (b) please? :) --- I have figured out (a) to be: a velocity of 2.92 m/s at 121 degrees (CCW from x axis). I figured it would take 805s to cross (the distance she travels, or the hypotonuse of the triangle, is 2333.27m). I really need help on part (b) though :(

Explanation / Answer

a) Your calculation of the total magnitude is correct; it's just an application of the Pythagorean theorem. Her angle is thus calculated using the formula, tangent(angle) = y/x. Your calculation is right (I got 120.96 degrees). However, as she's crossing the river, her speed is 2.5 m/s, because the direction of the current is perpendicular to her path. Thus the time it takes to cross is just 2000/2.5 = 800 s. b) For her to be traveling only in the y direction, the x axis velocity must be zero. So, her x-velocity must be +1.5 m/s (to cancel out the -1.5 m/s of the river). We know from the problem statement that her total speed is 2.5 m/s (this is the 'hypotenuse'). To find the angle, we use simple trigonometry. cos(angle) = adj/hyp angle = acos(1.5/2.5) = 53.13 degrees from the positive x-axis. Then, her speed across the width of the river is hyp*sin(angle) = 2.5*sin(53.13) = 2 m/s. Finally, to find how long it takes, we divide 2000 m / 2 m/s = 1000 s. The key to part (b) was the fact she was travelling straight up, so her x velocity had to be zero overall.

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