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Chap 03 Solutions copy

Chap 03 Solutions copy - VECTORS AND MOTION IN TWO...

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3-1 V ECTORS AND M OTION IN T WO D IMENSIONS 3 Q3.1. Reason: (a) If one component of the vector is zero, then the other component must not be zero (unless the whole vector is zero). Thus the magnitude of the vector will be the value of the other component. For example, if A x = 0 m and A y = 5 m, then the magnitude of the vector is A = (0 m) 2 + (5 m) 2 = 5 m (b) A zero magnitude says that the length of the vector is zero, thus each component must be zero. Assess: It stands to reason that a vector can have a nonzero magnitude with one component zero as long as the other one isn’t. It also makes sense that for the magnitude of the vector to be zero all the components must be zero. Q3.2. Reason: No, it is not possible. A scalar has a magnitude only but a vector has direction as well. Even if each has the same dimensions, the result of the addition of a scalar and a vector is ambiguous. Assess: We already learned in chapter 1 that we can’t add quantities unless they have the same dimensions; here we also point out that two quantities that will be added (or subtracted) must both be scalars or both vectors. Q3.3. Reason: Consider two vectors ! A and ! B . Their sum can be found using the method of algebraic addition. In Question 3.2 we found that the components of the zero vector are both zero. The components of the resultant of ! A and ! B must then be zero also. So R x = A x + B x = 0 R y = A y + B y = 0 Solving for the components of ! B in terms of ! A gives B x = ! A x and B y = ! A y . Then the magnitude of ! B is ( B x ) 2 + ( B y ) 2 = ( ! A x ) 2 + ( ! A y ) 2 = ( A x ) 2 + ( A y ) 2 . So then the magnitude of ! B is exactly equal to the magnitude of ! A . Assess: For two vectors to add to zero, the vectors must have exactly the same magnitude and point in opposite directions. Q3.4. Reason: (a) C = A + B only if ! A and ! B are in the same direction. Size does not matter. (b) C = A B if ! A and ! B are in the opposite direction to each other. Size matters only in that A > B because C as a magnitude can only be positive. Assess: Visualize the situation with arrows.
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3-2 Chapter 3 Q3.5. Reason: The ones that are constant are v x , a x , and a y . Furthermore, a x is not only constant, it is zero. Assess: There are instants when other quantities can be zero, but not throughout the flight. Remember that a y = g throughout the flight and that v x is constant; that is, projectile motion is nothing more than the combination of two simple kinds of motion: constant horizontal velocity and constant vertical acceleration. Q3.6. Reason: The acceleration of the ball is due to gravity, so the acceleration the ball experiences is always straight downward. (a) The velocity vector of the ball always has a component in the horizontal direction since it was thrown at an angle of 40 ° . The horizontal component of the ball’s velocity is constant throughout its trajectory. Since the velocity vector always has a component in the horizontal direction, it is never pointing entirely straight up or down.
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