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Q9) Figure 4-26 shows three paths for a football kicked from ground level. Ignoring the effects of air, rank the paths according to (a) time of flight, (b) initial vertical velocity component, (c) initial horizontal velocity component, and (d) initial speed, greatest first. p) •3 A positron undergoes a displacement Δr → = 2.0iˆ − 3.0jˆ + 6.0kˆ , ending with the position vector r → = 3.0jˆ − 4.0kˆ , in meters. What was the positron’s initial position vector? •5 SSM A train at a constant 60.0 km/h moves east for 40.0 min, then in a direction 50.0° east of due north for 20.0 min, and then west for 50.0 min. What are the (a) magnitude and (b) angle of its average velocity during this trip? •7 An ion’s position vector is initially r → = 5.0iˆ − 6.0jˆ + 2.0kˆ , and 10 s later it is r → = −2.0iˆ + 8.0jˆ − 2.0kˆ , all in meters. In unit-vector notation, what is its v → avg during the 10 s? •11 The position r → of a particle moving in an xy plane is given by r → = (2.00t3 − 5.00t)iˆ + (6.00 − 7.00t4)jˆ, with r → in meters and t in seconds. In unit-vector notation, calculate (a) r →, (b) v →, and (c) a → for t = 2.00 s. (d) What is the angle between the positive direction of the x axis and a line tangent to the particle’s path at t = 2.00 s? •13 SSM A particle moves so that its position (in meters) as a function of time (in seconds) is r → = iˆ + 4t2jˆ + tkˆ . Write expressions for (a) its velocity and (b) its acceleration as functions of time. •21 A dart is thrown horizontally with an initial speed of 10 m/s toward point P, the bull’s-eye on a dart board. It hits at point Q on the rim, vertically below P, 0.19 s later. (a) What is the distance PQ? (b) How far away from the dart board is the dart released? •22 A small ball rolls horizontally off the edge of a tabletop that is 1.20 m high. It strikes the floor at a point 1.52 m horizontally from the table edge. (a) How long is the ball in the air? (b) What is its speed at the instant it leaves the table? ••29 A projectile’s launch speed is five times its speed at maximum height. Find launch angle θ0. ••35 SSM A rifle that shoots bullets at 460 m/s is to be aimed at a target 45.7 m away. If the center of the target is level with the rifle, how high above the target must the rifle barrel be pointed so that the bullet hits dead center? ••37 SSM WWW A lowly high diver pushes off horizontally with a speed of 2.00 m/s from the platform edge 10.0 m above the surface of the water. (a) At what horizontal distance from the edge is the diver 0.800 s after pushing off? (b) At what vertical distance above the surface of the water is the diver just then? (c) At what horizontal distance from the edge does the diver strike the water? •••53 In Fig. 4-44, a baseball is hit at a height h = 1.00 m and then caught at the same height. It travels alongside a wall, moving up past the top of the wall 1.00 s after it is hit and then down past the top of the wall 4.00 s later, at distance D = 50.0 m farther along the wall. (a) What horizontal distance is traveled by the ball from hit to catch? What are the (b) magnitude and (c) angle (relative to the horizontal) of the ball’s velocity just after being hit? (d) How high is the wall? •62 What is the magnitude of the acceleration of a sprinter running at 10 m/s when rounding a turn of radius 25 m? part 2: 4 Equation 3-2 shows that the addition of two vectors a → and b → is commutative. Does that mean subtraction is commutative, so that 5 Which of the arrangements of axes in Fig. 3-23 can be labeled “right-handed coordinate system”? As usual, each axis label indicates the positive side of the axis. 8 If a → ⋅ b → = a → ⋅ c →, must b → equal c →? 9 If F → = q(v → × B → ) and v → is perpendicular to B → , then what is the direction of B → in the three situations shown in Fig. 3-24 when constant q is (a) positive and (b) negative? Problems: •1 SSM What are (a) the x component and (b) the y component of a vector a → in the xy plane if its direction is 250° counterclockwise from the positive direction of the x axis and its magnitude is 7.3 m? •3 SSM The x component of vector A → is ‒25.0 m and the y component is +40.0 m. (a) What is the magnitude of A → ? (b) What is the angle between the direction of A → and the positive direction of x? •5 A ship sets out to sail to a point 120 km due north. An unexpected storm blows the ship to a point 100 km due east of its starting point. (a) How far and (b) in what direction must it now sail to reach its original destination? •9 Two vectors are given by a→ = (4.0 m)iˆ ‒ (3.0 m)jˆ + (1.0 m)kˆ and b → = (‒1.0 m)iˆ + (1.0 m)jˆ + (4.0 m)kˆ . In unit-vector notation, find (a) a → + b → , (b) a → ‒ b → , and (c) a third vector c → such that a → ‒ b → + c → = 0. •11 SSM (a) In unit-vector notation, what is the sum a → + b → if a → = (4.0 m)iˆ + (3.0 m)kˆ and b → = (‒13.0 m)iˆ + (7.0 m)kˆ ? What are the (b) magnitude and (c) direction of a → + b → ? •13 A person desires to reach a point that is 3.40 km from her present location and in a direction that is 35.0° north of east. However, she must travel along streets that are oriented either north– south or east–west. What is the minimum distance she could travel to reach her destination? •17 ILW Three vectors a →, b → , and c → each have a magnitude of 50 m and lie in an xy plane. Their directions relative to the positive direction of the x axis are 30°, 195°, and 315°, respectively. What are (a) the magnitude and (b) the angle of the vector a → + b → + c →, and (c) the magnitude and (d) the angle of a → ‒ b → + c →? What are the (e) magnitude and (f) angle of a fourth vector d → such that (a → + b → ) − (c → + d → ) = 0? •33 For the vectors in Fig. 3-32, with a = 4, b = 3, and c = 5, what are (a) the magnitude and (b) the direction of a → × b → , (c) the magnitude and (d) the direction of a → × c →, and (e) the magnitude and (f) the direction of b→ × c →? (The z axis is not shown.) •34 Two vectors are presented as a→ = 3.0iˆ + 5.0jˆ and b → = 2.0iˆ + 4.0jˆ. Find (a) a → × b → , (b) a → · b → , (c) (a → + b →) · b → , and (d) the component of a → along the direction of b→ . •37 Three vectors are given by a → = 3.0iˆ + 3.0jˆ − 2.0kˆ , b→ = −1.0iˆ − 4.0jˆ + 2.0kˆ , and c → = 2.0iˆ + 2.0jˆ + 1.0kˆ . Find (a) a → · (b → × c →) (b) a → · (b → + c →), and (c) a → × (b → + c →). 米米米 关于十 (d) S. Figure 3-23 Question 5. 北平 (3) Figure 3-24 Question 9. 7 -X Figure 3-32 Problems 33 and 54. Figure 4-26 Question 9.
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