I am having some problems with the first problem (2/201)
first I tried to break all of the numbers into the relative velocity vectors and obtained the following.
Vb = 60 i
Va = Va (cos 20 i + sin 20 j)
Va/b = Va/b( cos (theta) i + sin (theta) j)
Then i tried to break up these equations and make an i equation and a j equation based on
Va = Vb - Va/b
this yielded
60i = Va cos 20 i + Va/b cos (theta) i
0 = Va (sin (20) j +Va/b sin (theta) j
From here I was left with three variables and two equations... am I missing something?
6 comments:
at a glance, all i noticed was that your relative v equation wasn't quite right.
try:
Va/b = Va - Vb
in addition to what i said before.. since you are using the angle with the x-axis for Va, "theta" should be 30. i'm not sure if your work comes out right, but if you drew your V vector triangle right, it should work.
An important point in this problem is that, since the tow rope does not stretch, the velocity of A relative to B must be perpendicular to the rope. Therefore, the direction of the relative velocity vector v_A/B has x-y components as shown in the figure (I attached a figure to the original post.)
Therefore, the direction of v_A/B is known in terms of the angle theta as shown.
Let me know if this does not help.
i understand that the picture is not necessarily to scale, but shouldn't Va/b be the vector difference Va - Vb? if that's the case, it seems to me Va/b would not be perpendicular to the rope.
can you explain in a different way why Va/b must be perpendicular to the rope (maybe in class)?
There are a couple arguments that can be made.
One way is to have a polar coordinate system corresponding to someone at the rear of B watching A: e_r points toward A away from B and e_theta is perpendicular to the rope. Let R be the length of the tow rope. If we are able to assume that the rope does not go slack, then the distance R between A and B cannot change. Therefore, R_dot = 0, and as a result v_A/B has only a component in e_theta which is perpendicular to the tow ropes. (This is a good perspective to use since the second part of the question asks for theta_dot -- this comes directly from the polar coordinate perspective.)
From a slightly different perspective. Imagine yourself on boat B watching A. You cannot observe any motion of A directly toward you; otherwise the rope goes slack. Therefore the only motion that you see of A must be perpendicular to the tow rope.
I see your point in that it is hard to see that v_A - v_B is perpendicular to the rope (since the scale of my drawing is not great). Once you get your final answer, draw an accurate figure. It will make sense then, I think.
Good way to check things out -- make sure that all of this makes sense before moving on. Ask about this in the problem-solving session on Friday.
thank you, i think it's clear now.
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