LAB 05 Projectile Motion

Transcription

1 LAB 5 Projectile Motion CONTENT: 1. Introduction. Projectile motion A. Setup B. Various characteristics 3. Pre-lab: A. Activities B. Preliminar info C. Quiz 1. Introduction After introducing one-dimensional motion, the net step in our stud of motion is to allow bodies to move in a plane that is, perform two-dimensional motions. In this case, taking advantage from the propert of vectors is crucial, inasmuch as kinematic quantities such as position, velocit, and acceleration can be represented as arrows ling in the same velocit plane, as depicted in the adjacent figure where I denoted r and respectivel v the position and the velocit of a car in the -plane of the road. Note that, in a given coordinate sstem, the vector position is an arrow connecting the origin to the r location of the object, while the velocit is an arrow pointing alwas in the direction of motion, such that it stas everwhere tangent to the trajector of the object. In turn, acceleration if an will have the direction of whatever net force acts on the object, changing its velocit. In our class, we illustrate the vector-based strateg to model two-dimensional motions using an eample: the trajector of a projectile in its simplest variant, when it is overwhelmingl determined b weight. v. Projectile motion Let us consider a projectile launched with a certain muzzle velocit (tpicall called initial velocit). How can we understand its ulterior trajector in phsical terms? Note that, if we neglect air resistance (a.k.a. drag), as soon as the object leaves the launcher the onl pull eerted on it is its own vertical weight. Hence, its velocit will tend to rotate to point downwards. Yet, this cannot happen instantaneousl: the velocit will change direction graduall, such that the projectile will move along its characteristicall vaulted trajector. Since the cause of this change of direction is gravit, it is given b gravitational acceleration. With a modicum of curiosit, ou ma summon what we learned in class about vectors to observe how gravitational acceleration g is given b the change of velocit v v 1 between an two points on the trajector divided b the traveling time Δt, as shown in the diagram. To set off the analsis of projectile motion, one needs to reformulate the question that canalized out discussions about kinematics: Knowing the initial position and velocit of the projectile, how can one predict its position and velocit at a later time? In the past, we answered this question for some one-dimensional motions b developing kinematic equations. Recall for instance the equations for uniforml accelerated motion which we applied to the vertical free fall. In turn, to answer the question in two-dimensional situations, one needs to emplo vector properties more methodicall. v 1 Δv v v 1 g = Δv Δt = v v 1 Δt 1

2 A. Setup The step-b-step strateg was introduced in class: 1. Embed the two-dimensional trajector in a coordinate sstem, for instance including horizontal and vertical aes, with the projectile initiall at coordinates and.. Split the initial velocit v of the projectile into - and -components, v and v,which can be calculated in terms of initial speed v and initial launching angle θ : v v cos (1) v v sin () 3. Note that if onl weight acts on the projectile the acceleration is verticall downward, so the -component of the velocit stas constant while the -component is accelerated uniforml. So, we can calculate the components of the position and velocit at an later time t using the standard equations with zero and then constant acceleration g: Along : vt (3) v v (4) Along : (5) v t gt v v gt (6) 4. Once the components at the later time t are found, one can calculate the two-dimensional velocit and position using the formulas relating the magnitude and direction of a vector to its components. For instance, for the velocit at time t: Magnitude: 1 Direction: tan v v v v v (7) (8) Eample: A projectile is launched with initial speed v = 98 m s at an angle θ = 3 above the horizontal, as shown on the figure. Neglecting air resistance, what are the coordinates and the velocit of the projectile 8seconds later? (m) ma (m) θ R 9 initial, =, = later time t The components of the initial velocit are: v 98 m/scos3 85 m/s v 98 m/s sin 3 49 m/s Consequentl, using equations (3)-(6), the coordinates and velocit are given b v t 85 m/s 8. s 68 m v t gt 78.4 m 85 m/s v v v v v 85 m/s 9 m/s 87 m/s v 9 m/s v gt 49 m/s 9.8 m/s 8. s 9 m/s tan m/s

3 3. Pre-lab A. Activities 1. Read carefull the introductor material provided above. Make sure that ou understand: The principles that shape up the trajector of a projectile. The logic of using the equations along horizontal and vertical aes to predict the coordinates and velocit of the projectile at an time after it is launched.. Answer the questions on the quiz at the end of this document. Note that the are based on observing a simulation of projectile motion accessible via the course homepage. The questions are also available on the Blackboard site associated with the PHYS 17 lecture. B. Preliminar information The subject of LAB 5 is Projectile Motion. Its main goal is to develop and help ou practice our analtical skills in studing the two-dimensional motion of a projectile. You are designing the specifications for a machine which must throw an object from an elevation (the lab table), across an area which ou do not have access to, to a safe area below (on the floor). To illustrate the principles of our machine, ou will make predictions about the trajector of the projectile using the kinematic model developed in class (also discussed above), and then emplo a ballistic pendulum as a projectile launcher to demonstrate the viabilit of the model b testing the predictions Technical Comments: You will tackle the eperiment in two steps: 1. PART 1: Familiarize ourselves with the launcher, and measure the muzzle velocit of projectiles.. PART : Use the initial position and velocit of the projectile to hit a target placed at a location predicted b the theoretical model. In none of the two parts ou will measure the time that the projectile spends in the air. So, the theoretical strateg will be to combine equations (3) and (5) above. That is, substitute the time from equation (5) to (3), thence obtain relationships between vertical and horizontal displacements. In PART 1, the projectile will be fired horizontall from a desk on the floor (θ = ). You will measure the horizontal and vertical displacements, and use them to estimate the initial velocit of the projectile. The necessar calculation was performed in class (Problem 3). Recall that we obtained the time from Eq. (5) for the vertical displacement Δ b setting v = and then substituting it in Eq. (3) for the horizontal displacement Δ. Review that calculation because ou will have to reproduce it on our lab report. In PART, the projectile will be fired with a non-zero initial angle from a desk on the floor (θ ). The initial velocit of the projectile will be assumed known (from PART 1) as well as the vertical displacement (measured). In turn, ou will have to predict and test the horizontal displacement Δ. However, to find the time t from Eq. (5) is a tad more challenging, because now v = v sin θ, so the time is a solution of the quadratic equation: v Δ (Eq.5) vsint gt (9) measure PART 1 set v θ Δ (Eq.3) This equation has two solutions, but one of them is negative, so ou will need onl the positive one which makes phsical sense: t flight v sin v sin g (1) g Δ (Eq.5) Δ (Eq.3) 3

4 Note that, when the projectile is fired at an initial altitude h, the vertical displacement is Δ = = h = h. Then, our theoretical horizontal displacement will be calculated from v cos t. (11) flight Understand and practice these calculations because ou will have to reproduce them on our lab report. N.B. This lab is a formal one, so even though still based on team work each student will have to turn in an individual lab report. (Quiz on the net page) 4

5 C. Quiz 5 Name: Instructions: The first three questions on the quiz require that ou observe the motion of a projectile in a simulation accessible using the link Enable Adobe Flash in order to see the animation. Click [CM] on the left-hand-side list, or scroll down to the animation dubbed CM showing a red projectile read to launch. The red arrows represent the vector velocit and its components. The instructions for using the buttons are on the left. Don t worr about the technicalities about the air resistance (drag), just think about it in qualitative terms as a force alwas acting against the direction of motion. Q1. [] Make sure that the [Drag] bo is unchecked. Click the [Start] button and observe the horizontal component v of the velocit. Which of the following is an eplanation for the behavior of this component? a) In the first half, the weight decelerates the projectile so v gets smaller and smaller. b) In the second half, the weight accelerates the projectile so v gets larger and larger. c) As it moves, the projectile loses momentum, so v gets smaller and smaller throughout. d) In the first half, the force delivered b the launcher cancels the weight, so v stas constant. e) There is no horizontal force acting on the projectile, so v stas constant throughout. Q. [] Make sure that the [Drag] bo is unchecked. Click the [Start] button and let the projectile complete a trajector. Then check the [Drag] bo. Click [Reset] to bring the projectile at start and launch it again. [Reset] and repeat the launch with the drag strengthened b increasing the k-constant to. Which of the following is an eplanation for the shapes of the three observed trajectories? a) All trajectories are left-right smmetric because the forces act in the same wa throughout. b) Onl the no-drag trajector is left-right smmetric, because there is no horizontal force on the projectile. c) The stronger is the drag, the taller the trajectories, because air resistance compresses them horizontall. d) The stronger is the drag, the shorter and asmmetric the trajectories, because air resistance decreases the - component of the velocit. e) Both (b) and (d) are true. Q3. [] Uncheck the [Drag] bo. Click [Rescale] to clear the screen. Repeat the projectile launch for the following initial angles: θ =5º, 35º, 45º, 55º, 65º (make sure that ever time ou change the angle in the bo, ou click in another bo for the new value to take in the vectors should redirect to follow the new angle). Which of the following is true about the observed trajectories? a) The larger the initial angle the shorter the trajectories. b) The same horizontal range can be reached for two initial angles: one lower and the other larger than 45º. c) The maimum range is reached when θ = 45º. d) The larger the initial angle, the longer is the time spent b the projectile in the air. e) Answers (b), (c) and (d) are all true. Q4. [] What is the main goal of our net lab eperiment? a) To measure the muzzle speed of a projectile. b) To measure the distance traveled b a projectile fired and landing at the same vertical level. c) To measure the distance traveled b a projectile fired horizontall from a desk and landing on the floor. d) To verif that the prediction made b the two-dimensional kinematic model for projectile motion is confirmed b the motion of an actual projectile. e) None of the above. Q5. [] Sa that in PART of our net lab our launcher is 1.1 meters above the floor. Your projectile is launched with an initial speed of 3.4 m/s, at an angle of 3º. What is the horizontal distance that ou should epect our projectile to travel before it touches the floor? (Hint: read and follow the Technical Comments in the pre-lab.) a) About meters b) About 1.8 meters c) About.7 meters d) About 1 meter e) None of the above. 5