PHYS 100 Discussion Session 3 2D Relative Motion Week 04 The Plan This week is about two main ideas, practicing vector addition and understanding relative motion. You ll accomplish both by looking at two relative motion situations. The first is the stream-crossing problem we talked about in lecture. You ll work with your group to find crossing times for three different angles of attack across the water. The second situation involves a swimmer caught in a riptide. The latter portions of this problem give you practice going between component and magnitude/direction representations of vectors. 1
A boat is trying to cross a flowing stream. The stream is 100 meters wide, the boat travels 5 m/s in still water, and the stream flows 1 m/s with respect to shore. DQ1) Let s say that the boat points 45 from shore. a. On the figure below, sketch the path that the boat takes from the reference frame of the shore. Compare your predicted path with the other people at your table. Do you all agree? b. Ultimately, we want to determine the time it takes the boat to cross the river. For the first step, we want to determine θbg, the angle of the boat s path, as measured from the x-axis, in the reference frame of the ground? Talk with your group to define some symbols that represent quantities that you are given or will need to solve this problem and fill in the table below Symbol Meaning of Symbol Value (if known) W vwg vbw vbg θwg θbw θbg TBG Width of river Velocity of the water wrt ground Velocity of boat wrt water Velocity of boat wrt ground Angle from x-axis of water wrt ground Angle from x-axis of boat wrt water Angle from x-axis of boat wrt ground Time it takes boat to cross river 100 m 1 m/s 5 m/s? 0 45 o?? In the space below write down the equation(s) using these symbols that will be necessary to solve this problem. / 2
Solve these equations to determine θbg, the angle of the boat s path as measured from the x-axis, in the reference frame of the ground. Putting in the numbers, tanθbg = 0.78 θbg = 37.9 o c. What is DBG, the total distance in the reference frame of the ground, that the boat travels? DBG = 163 m d. What is TBG, the time it takes the boat to cross the river? We do not know vbg, but could find it from (vbg)x and (vbg)y. The quicker way is to notice from the eequations at the top of the page: DBG = W/vBGsinθBG, and vbgsinθbg = vbwsinθbw TBG = 28.3 s 3
We will now solve this problem from the reference frame of the water. A boat is trying to cross a flowing stream. The stream is 100 meters wide, the boat travels 5 m/s in still water, and the stream flows 1 m/s with respect to shore. DQ2) Let s say that the boat points 45 from shore. a. On the figure below, sketch the path that the boat takes from the reference frame of the water. Compare your predicted path with the other people at your table. Do you all agree? b. Ultimately, we want to determine the time it takes the boat to cross the river. For the first step, we want to determine θbw, the angle of the boat s path as measured from the x-axis, in the reference frame of the water. The symbols you will need will be the set you used in DQ1, so you do not need to write them down again. In the space below write down the equation(s) using these symbols that will be necessary to solve this problem. / 4
Solve these equations to find an expression for θbw, the angle of the boat s path as measured from the x-axis, in the reference frame of the water. It was given!! θbw = 45 o c. What is DBW, the total distance in the reference frame of the water, that the boat travels? DBW = 141 m d. What is TBW, the time it takes the boat to cross the river? / TBW = 28.3 s 5
Reflecting Back: Finding Meaning a. Looking back at your calculations, you should see that the time it takes to cross the river was the same for each reference frame (as it must be). In which reference frame was the calculation easier? Why? I believe the calculation was much easier in the reference frame of the water because you are given the speed of the boat in the frame of the water and the distance in the water is easy to calculate. In the frame of the ground, you need to know the speed of the boat in the ground frame. To make this calculation, you need to use the relative motion equation and the resulting calculations are a bit messier. The time it takes to cross the river is always just the distance divided by the speed in that direction, in whatever frame it is easier to make the calculation. b. Write down the algebraic expression for the time it takes the boat to cross the river from the reference frame of the water calculation. / Can you see from this equation what angle the boat must point in order to cross the river in the least amount of time? Does your result agree with the swimmer problem we discussed in class? The time is inversely proportional to the sine of the angle. Therefore, the minimum time corresponds to the maximum of the sine. The maximum of the sine occurs at θbw = 90 o. 6
DQ3) A riptide is a strong current out into the open ocean. Riptides are formed when an underwater sandbar near shore has been broken. Receding water is directed by the sandbar to the hole and a strong isolated current away from shore is created. See diagram below: Underwater sandbar y x Shore (Beach) Talk to your group members about the following questions. Answer the following questions with both a picture of the vector asked for and numerical values for the magnitude and direction. Record your group consensus on the group answer sheet at your table. A swimmer is caught in the riptide. The swimmer can swim at a max speed of 1 m/s in still water. The speed of the riptide current is 2 m/s with respect to shore. The swimmer makes the common mistake of trying to swim directly toward the beach. (For the following problems use the axes drawn on the diagram to determine the positive directions.) a. From the perspective of someone on the beach, what are the velocities of the swimmer and the water? (Give speed and direction) 7
b. From the perspective of the swimmer what are the velocities of the water and the person on the beach? (Give speed and direction) 8
Underwater sandbar y x Shore (Beach) The person on the beach yells at the swimmer to swim parallel to the shore. So the swimmer swims in the positive x direction. The riptide remains at 2 m/s and the swimmer can swim at 1 m/s through still water. c. From the perspective of someone on the beach what are the velocities of the swimmer and the water? (Give speed and direction) 9
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d. From the perspective of the swimmer what are the velocities of the water and the person on the beach? (Give speed and direction) 11
Formula Sheet Definitions Position Velocity Acceleration x v = dx dt a = = dv dt 2 d x 2 dt Constant Acceleration v v at = 0 + 2 x = x + v t + at 1 0 0 2 ( ) v = v + 2a x x 2 2 0 0 Relative Motion va, B = va, E + ve, B v = v E, B B, E Constants and Conversions m g = 9. 81 = 32 ft 2 2 s s 1 mile = 1. 609 km Quadratic Formula If 2 2 ± 4 ax + bx + c = 0 then x = b b ac 2a 12