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Physics 114B - Mechanics Lecture 9 (Walker: 4.3-5) 2D Kinematics Examples January 21, 2014

Physics 114B - Mechanics Lecture 9 (Walker: 4.3-5) 2D Kinematics Examples January 21, 2014. John G. Cramer Professor Emeritus, Department of Physics B451 PAB jcramer@uw.edu. Announcements.

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Physics 114B - Mechanics Lecture 9 (Walker: 4.3-5) 2D Kinematics Examples January 21, 2014

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  1. Physics 114B - MechanicsLecture 9 (Walker: 4.3-5)2D Kinematics ExamplesJanuary 21, 2014 John G. Cramer Professor Emeritus, Department of Physics B451 PAB jcramer@uw.edu

  2. Announcements • Homework Assignments #2-4 are now available on WebAssign. Assignment #2 is due at 11:59 PM on Thursday, January 23. Assignment #3 is due at 11:59 PM on Thursday, January 30. • We will have our first Midterm Exam on Friday, January 24. It will have 75 points of multiple-choice questions based on end-of-chapter problems, and a 25 point free-response question testing qualitative understanding of Chapters 1-4. • On Thursday, we will have a pre-exam review. This is a good time to ask questions and ask for further explanations of the material that has been covered in the lectures. • Exam I will have assigned seating. Seat assignments will be posted before the exam on the entry doors. If you would like to request a left-handed seat, a right-handed aisle seat, a front-row seat, or need a seat at the tables in the back (4 seats only), send me E-mail ASAP. Physics 114A - Lecture 9

  3. Lecture Schedule (Part 1) We are here. Physics 114A - Lecture 9

  4. Finding Position from Velocity Physics 114A - Lecture 9

  5. Drag Racer’s Displacement The figure shows thevelocity of a drag racer.How far does the racermove during the first 3.0 s? Solution: The net distance traveledis the area under the velocitycurve shown in blue. This isa triangle with sides 12 m/sand 3.0 s. The area of thistriangle is: A = ½(12 m/s)(3 s) = 18 m. Thus, the drag racer moves18 m in the first 3 seconds. Physics 114A - Lecture 9

  6. Zero Launch Angle Launch angle: direction of initial velocity with respect to horizontal Physics 114A - Lecture 9

  7. Zero Launch Angle In the zero launch angle case, the initial velocity in the y-direction is zero. Here are the equations of motion, with x0 = 0 and y0 = h: Physics 114A - Lecture 9

  8. Zero-Launch Trajectory This is the trajectory of a projectile launched horizontally. It is a parabola. Physics 114A - Lecture 9

  9. Zero Launch Angle Eliminating t and solving for y as a function of x: This has the form y = a + bx2, which is the equation of a parabola. The landing point can be found by setting y = 0 and solving for x: Physics 114A - Lecture 9

  10. g Projectile Motion This is the equationfor a parabola. Physics 114A - Lecture 9

  11. General Launch Angle In general, v0x = v0 cos θand v0y = v0 sin θ This gives the equations of motion: Physics 114A - Lecture 9

  12. Clicker Question 1 You throw a ball into the air with an initial speed of 10 m/s at an angle of 60° above the horizontal. The ball returns to the height at which it was thrown in a time T. Which of these plots best represents the speed of the ball as a function of time? Physics 114A - Lecture 9

  13. Range General Launch Angle Snapshots of a trajectory; red dots are at t = 1 s, t = 2 s, and t = 3 s Physics 114A - Lecture 9

  14. Projectile Motion: Range Range: the horizontal distance a projectile travels If the initial and final elevation are the same: Physics 114A - Lecture 9

  15. Projectile Motion:Maximum Range The range is a maximum when θ = 45°: Physics 114A - Lecture 9

  16. Projectile Motion:Symmetry Symmetry in projectile motion: Same y and |v| Same y and |v| Physics 114A - Lecture 9

  17. y x Path of Projectile Physics 114A - Lecture 9

  18. g Example: A Cap in the Air A delighted physics graduate throws her cap into the air with an initial velocity of 24.5 m/s at a 36.9° angle with the horizontal. The cap is later caught by another student. [Neglect air resistance ; Sin(36.9°) = 3/5; Cos(36.9°) = 4/5]. (a) Find the total time the cap is in the air.(b) Find the total horizontal distance traveled. Physics 114A - Lecture 9

  19. Example: Ranger & Monkey A park ranger with a tranquilizer dart intends to shoot amonkey hanging from a branch. The ranger points the barreldirectly at the monkey, not realizing that the dart willfollow a parabolic path that will pass below themonkey’s present position. The monkey,seeing the gun discharge, instantly lets go of the branch and drops out of thetree, expecting to avoid the dart. (a) Show that the monkey will be hit,independent of the dart velocity vd.(b) Let vd0 be the initial velocity of the dart. Find the dart velocity relative to the monkey at an arbitrary time t during the dart’s flight. r = 0 In other words, provided the gun is aimed so that the extension of thevector vdg0 passes through the monkey’s position rm0, the dart hits the monkey. Physics 114A - Lecture 9

  20. g Example: A Supply Drop A helicopter drops a supply package to flood victims on a raft in a swollen lake. When the package is released, the helicopter is 100 m directly above the raft and flying at a velocity of 25.0 m/s at an angle of 36.9° above the horizontal. [Neglect air resistance; Sin(36.9°) = 3/5; Cos(36.9°) = 4/5].(a) How long is the package in the air?(b) How far from the raft does the package land? (c) If the helicopter continues at constant speed, where is it when the package lands? The helicopter will be 194 m above the package impact point. Physics 114A - Lecture 9

  21. Example: A Rocket Sled A rocket sled acceleratesat 50 m/s2 for 5.0 s, coastsfor 3.0 s, then deploys abraking parachute anddecelerates at 3.0 m/s2until coming to a halt. • What is the maximumvelocity reached bythe rocket sled? • v1x = v0x + a0x(t1 – t0)= a0xt1 • = (50 m/s2)(5.0 s)= 250 m/s (b) What is the total distance traveled bythe rocket sled? x1 = x0 + v0x(t1 – t0) + ½a0x(t1 – t0)2 = ½a0xt12 = ½(50 m/s2)(5.0 s)2 = 625 m x2 = x1 + v1x(t2 – t1) = 625 m + (250 m/s)(3.0 s) = 1375 m v3x2 = 0 = v2x2 +2a2xDx = v2x2 +2a2x(x3 – x2), so x3 = x2 + (v3x2 - v2x2)/2a2x = 1375 m + [0 – (250 m/s)2]/[2(-3.0 m/s2)] = 11,800 m Physics 114A - Lecture 9

  22. Example: Dropping a Ball A person skateboarding witha constant speed of 1.30 m/sreleases a ball from a heightof 1.25 m above the ground.Use x0 = 0 and y0 = h = 1.25 m. (a) Find x and y for t1 = 0.250 s. (b) Find x and y for t2 = 0.500 s. (c) Find the velocity, speed, and direction of the ball at t2 = 0.500 s. Physics 114A - Lecture 9

  23. Clicker Question 2:Dropping the Ball From the sameheight (and at the sametime), one ball is dropped and another ball is fired horizontally. Which one will hit the ground first? (A) the “dropped” ball (B) the “fired” ball (C) they both hit at the same time (D) it depends on how hard the ball was fired (E) it depends on the initial height Physics 114A - Lecture 9

  24. t1 Example: A Rough Shot Chipping from the rough, a golfer sends the ball over a 3.0 m high tree that is 14.0 m away. The ball lands on the green at the same level from which it was struck after traveling a horizontal distance of 17.8 m. (a) If the ball left the clubat 54.0° above the horizontaland landed 2.24 s later, what was its initial speed v0? (b) How high was the ball when it passed over the tree? Physics 114A - Lecture 9

  25. g Example: A Home Run A baseball is hit so that it leaves the bat making a 30° angle with the ground. It crosses a low fence at the boundary of the ballpark 100 m from home plate at the same height that it was struck. (Neglect air resistance.) What was its velocity as it left the bat? Physics 114A - Lecture 9

  26. Clicker Question 3:War at Sea 1 2 A battleship simultaneously fires two shells at two enemy submarines. The shells are launched with the same initial velocity. If the shells follow the trajectories shown, which submarine gets hit first ? (A) Submarine 1 (B) Submarine 2 (C) They are both hit at the same time (D) It depends on the initial velocity Physics 114A - Lecture 9

  27. g R is maximum when qo=45o, so that sin 2qo = 1. Horizontal Range of a Projectile Physics 114A - Lecture 9

  28. Example: An Elevated Green A golfer hits a ball from the origin with an initial speed of 30.0 m/s at an angle of 50.0° above the horizontal. The ball lands on a green that is 5.00 m above the level where the ball was struck. • (a) How long was the ball in the air? • How far did the ball travel horizontally before ot landed? • What is the speed and direction of the ball just before it lands? Physics 114A - Lecture 9

  29. Example: To Catch a Thief A police officer chases a master jewelthief across city rooftops. They are bothrunning when they come to a gap betweenbuildings that is 4.0 m wide and has a dropof 3.0 m. The thief having studied a littlephysics, leaps at 5.0 m/s at an angle of 45° above the horizontal and clears the gap easily. The police officer did not study physics and thinks he should maximize his horizontal velocity, so he leaps horizontally at 5.0 m/s. (a) Does he clear the gap?(b) By how much does the thief clear the gap? No! Physics 114A - Lecture 9

  30. Example: What A Shot! The archerfish hunts by dislodging an unsuspecting insect from its resting place with a stream of water expelled from the fish’s mouth. Suppose an archerfish squirts water with an initial speed of 2.30 m/s at an angle of 19.5° above the horizontal. When the water reaches a beetle ata height h above the water’ssurface, it is moving horizontally. • (a) How much time does the beetle have to react? • What is the height h of the beetle? • What is the horizontal distance d between the fish and the beetle? Physics 114A - Lecture 9

  31. v0=7.64 m/s q = 52.5° v0 q Example: Basketball Throw A basketball player stands 15.0 ft from the basket, which is 10.0 ft above the floor. The ball leaves his hands 6.0 ft above the floor with an initial velocity v0 that is directed at an angle q with the horizontal. What are the conditions on v0 and q, such that he makes the basket? Physics 114A - Lecture 9

  32. Summary - 2D Kinematics • Components of motion in the x- and y-directions can be treated independently. • In projectile motion, the acceleration is –g. • If the launch angle is zero, the initial velocity has only an x-component. • The path followed by a projectile is a parabola. • The range is the horizontal distance the projectile travels. Physics 114A - Lecture 9

  33. Homework Assignment #2 should be submitted using the Tycho system by11:59 PM on January 23. • We will have our first Midterm Exam on Friday, January 24. On Thursday, we will have a pre-exam review. Be prepared to ask questions about things that need clarification. End of Lecture 9

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