Motion in Two and Three Dimensions

Transcription

1 chapter 4 Motion in Two and Three Dimensions Projectile motion (Section 4.3) 1. Which target got hit first? Contet of the tetbook: Before Eample Projectile range problem comparable to Eample 7, ecept here the initial values given are v 0 and v 0y. 3. Gun and target 4. Gun and target on the moon Questions 4.3 and 4.4 are supplementary problems for projectial motion. Motion along a circular arc: These two questions are conventional problems in other engineering physics tetbooks beyond the uniform circular motion discussed in this section. They may be introduced after Eample 9 of Section Deceleration of a train 6. Motion of a simple pendulum Relative velocity and the addition of velocities. These questions may be used before Eample 10 in Section Definition of relative velocity basic conceptual question. 8. Crossing a river simplified version of Eample 10. Contents PhysiQuiz 29

2 y O υ 0 ' ' υ 0 h B Two projectiles are launched simultaneously and with the same launching speed from point O as shown v 0 v 0. The first is launched at angle and hits target in time t. The second is launched at a greater angle and hits target B in time t B. Compare the flight times for the two projectiles: B C t B t t B t t B t Eplanation: Consider the projected motion along the vertical direction. The flight time for the first case is given by t flight 2t rise 2v 0y /g. For the second case, the y component of the initial velocity v 0y v 0 sin is replaced by v 0y v 0 sin, where and v 0 v 0 are given. Therefore v 0y v 0y, which means that the flight of the second case takes longer. nswer. 30 PhysiQuiz 1. Which Target Got Hit First?

3 υ 0y y υ 0 υ 0 R h Given v 0 20 m/s and v 0y 10 m/s and g 10 m/s 2. Find R. Choose one of the following: B C 2v 0 2 /g 2( )/ m v 0 v 0y /g 20 10/10 20 m 2v 0 v 0y /g /10 40 m Hint: The rising time t rise v 0y /g, v 0 2 v 0 2 v 0y m 2. Etra: Keep v 0 fied and vary to find the maimum value of R. Eplanation: R v 0 (2t rise ) 2v 0 v 0y /g. nswer C Eplanation etra: R 2v 0 2 sin cos /g v 0 2 sin 2 /g. So the maimum value of R when v 0 2 is fied is R v 0 2 /g 50 m. 2. Projectile Range PhysiQuiz 31

4 y v 0 P υ 0y O υ 0 B Consider a gun at O aiming at a target located at point. The gun fires at time t 0, and at the same time the target begins to fall from rest. Let v O 10 m/s and v Oy 20 m/s. lso, let OB 20 m and B 40 m. Find the height BP as the bullet passes the vertical line B: B C BP (in m) Hint: In the direction t OP OB/v o 2 s. To find BP, use y v Oy t gt 2 /2 for the motion of the bullet along the y direction. ssume g 10 m/s 2. Eplanation: From point O to point P, the time that the bullet spends traveling vertically is the same as the time it spends traveling from O to P, t BP t OP 2 s. Use the equation of motion in the vertical direction: BP v Oy t BP gt BP 2 /2 (20)(2) (10/2)(2) 2 20 m. nswer B. n alternative method: BP may also be determined by considering the fall of the target. P (1/2)gt BP 2 (10/2)(2) 2 20 m. BP B P m. 32 PhysiQuiz 3. Gun and Target

5 y v 0 P υ 0y O υ 0 B Consider the gun and target setup of the previous problem, where the initial velocity vector v 0, the length of the base line OB, and the height B are kept fied. Compare the height BP, when the eperiment is carried out on Earth to the height BP, when the eperiment is done on the Moon: B C BP BP BP BP BP BP Eplanation: Because v 0 and OB are the same for the two cases, the time of flight t OB/v 0 is also the same. During this time, on Earth the target falls by a distance P (1/2)gt 2. Because on the Moon the gravitational acceleration g is less than g, P P. In turn, BP BP. nswer C. 4. Gun and Target on the Moon PhysiQuiz 33

6 v O r Quadrant: II III I IV train is moving along a circular track of radius r 100 m. t point, v v 10 m/s. It is slowing down with a tangential deceleration of a magnitude a tangent a tangent 1 m/s 2. Sketch a total at. Which quadrant should it be in? B C D Quadrant I II III IV Etra: Find the magnitude a total. Eplanation: From the sketch at point, a total is in the third quadrant. nswer C. Eplanation etra: t point, centripetal acceleration a c v 2 /r 10 2 /100 1 m/s 2. So a total 2 a 2 c a 2 tangen t m/s 2, or 1.4 m/s 2. a centripetal a total a tangent 34 PhysiQuiz 5. Deceleration of a Train

7 g r v 0 g a φ a r a t simple pendulum consists of a string of length r and a ball attached to its end. When the string makes an angle with the vertical and the tangential velocity of the ball is pointing toward the vertical line, determine the corresponding tangential acceleration: B C D a tangent g sin g cos g sin g cos Hint: The angle is measured from the vertical line in a counterclockwise manner. Etra: Compare with in the sketch. Eplanation: From the sketch, we see that the tangential acceleration is pointing toward the vertical line; thus it has a sign opposite to that of. nswer C. Eplanation etra: Notice in the top sketch, the centrepetal vector a t g r where g r is shown in the second sketch. Since both vectors a and g are diagonals of two rectangles with same sides a t and g r,. One can see this equality visually, if a t is drawn to scale. 6. Motion of a Simple Pendulum PhysiQuiz 35 g q a t

8 30 mph 10 mph B Car travels at a speed of 30 mph to the right (positive direction) and car B travels at 10 mph to the left. Consider the velocity v B v B i to be the velocity of car observed by the driver in car B (in other words, v B is the velocity of relative to B). Given the ais orientation as shown, find v B and v B. Choose one (in mph): B C D v B v B Eplanation: Notice that the driver of car B sees that car is moving toward him or her that is, along the positive direction with a speed greater than v 30 mph. The algebra involved is given here: v B v v B 30 ( 10) 40 mph. v B v B v v B 40 mph. nswer D. 36 PhysiQuiz 7. Definition of Relative Velocity

9 v we v bw The diagram here shows a boat attempting to cross a river. ssume that the boat s speed relative to the water v bw 10 m/s and that the current (the water s speed relative to the Earth) v we 5 m/s. Find such that the boat crosses the river at a right angle to the bank: B C Etra: What is v be, the boat s speed relative to the Earth? Eplanation: Because sin v we /v bw 5/10 0.5, 30. Eplanation etra: v be v bw cos m/s. 8. Crossing a River PhysiQuiz 37