86_PartUniversity Physics Solution

86_PartUniversity Physics Solution - 3-16 Chapter 3...

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3-16 Chapter 3 EVALUATE: To travel due west the velocity of the plane relative to the air must have a westward component and also a component that is northward, opposite to the wind direction. Figure 3.40 3.41. IDENTIFY: Relative velocity problem in two dimensions. His motion relative to the earth (time displacement) depends on his velocity relative to the earth so we must solve for this velocity. (a) SET UP: View the motion from above. The velocity vectors in the problem are: M/E , v ! the velocity of the man relative to the earth W/E , v ! the velocity of the water relative to the earth M/W , v ! the velocity of the man relative to the water The rule for adding these velocities is M/E M/W W/E v= v + v ! !! Figure 3.41a The problem tells us that W/E v ! has magnitude 2.0 m/s and direction due south. It also tells us that M/W v ! has magnitude 4.2 m/s and direction due east. The vector addition diagram is then as shown in Figure 3.41b This diagram shows the vector addition M/E M/W W/E v ! and also has M/W v ! and W/E v ! in their specified directions. Note that the vector diagram forms a right triangle. Figure 3.41b The Pythagorean theorem applied to the vector addition diagram gives 22 2 M/E M/W W/E . vv v =+ EXECUTE: 22 2 2 M/E M/W W/E (4.2 m/s) (2.0 m/s) 4.7 m/s vv v = + = M/W W/E 4.2 m/s tan 2.10; 2.0 m/s v v θ == = 65 ; = ° or 90 25 . φ −=° The velocity of the man relative to the earth has magnitude 4.7 m/s and direction 25 S ° of E. (b) This requires careful thought. To cross the river the man must travel 800 m due east relative to the earth. The man&s velocity relative to the earth is M/E . v ! But, from the vector addition diagram the eastward component of M/E v equals M/W 4.2 m/s. v = Thus 0 800 m 190 s. 4.2 m/s x xx t v = (c) The southward component of M/E v ! equals W/E 2.0 m/s. v = Therefore, in the 190 s it takes him to cross the river the distance south the man travels relative to the earth is 0 (2.0 m/s)(190 s) 380 m. y yy v t −= = = EVALUATE: If there were no current he would cross in the same time, (800 m)/(4.2 m/s) 190 s. = The current carries him downstream but doesn&t affect his motion in the perpendicular direction, from bank to bank. 3.42. IDENTIFY: Use the relation that relates the relative velocities. SET UP: The relative velocities are the water relative to the earth, W/E v ! , the boat relative to the water, B/W v ! , and the boat relative to the earth, B/E v ! . B/E v ! is due east, W/E v ! is due south and has magnitude 2.0 m/s. B/W 4.2 m/s v = . B/E B/W W/E !! ! . The velocity addition diagram is given in Figure 3.42.
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Motion in Two or Three Dimensions 3-17 EXECUTE: (a) Find the direction of B/W v !
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86_PartUniversity Physics Solution - 3-16 Chapter 3...

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