Purpose of the experiment

Transcription

1 Projectile Motion PES 116 Advanced Physics Lab I Purpose of the experiment Measure the velocity of a ball using two photogates and Logger Pro. Apply the concepts of two-dimensional kinematics to predict the flight of a ball in projectile motion. FYI FYI It was discovered on a space mission that a frog can throw up. The frog throws up it stomach first, so the frog s stomach is dangling out of its mouth. Then the frog uses its forearms to dig out all the stomach s contents and then swallows the stomach back down again. Projectile Motion - 1

2 Table of Contents Background 3 Equipment 4 Time-of-Flight pad 7 Logger Pro 8 Procedure 9 Equipment List Logger Pro Pasco mini-launcher w/ mounting bracket Meter stick C clamp Photogates (x) Photogate mounting bracket Time-of-Flight pad w/ adapter Steel ball Plum bob Safety glasses (in box at front of room) Plunger Catch box carbon paper/ impact tape Projectile Motion -

3 Background If you have ever thrown a ball to another person, then you are an expert on twodimensional projectile motion. You know from the last lab that gravity pulls all objects down toward the center of the earth with an acceleration equal to g. Problem: you would like to throw a ball to your friend who is, let s say, 0 feet away. Everyone quickly learns that you cannot throw a ball in a straight line. You have to overcome the pull of gravity long enough so that the ball will reach your friend and not hit the ground. The solution: you have to throw the ball at an upward angle, giving the ball some motion in the x direction (toward your friend) and some motion in the y direction (up) to overcome gravity. The result is that the ball will travel along a parabolic path. When you throw a ball some of the velocity will be directed up (y direction) to overcome the pull of gravity. Gravity will constantly slow down the ball s upward travel until it brings it to a stop. The ball will then fall back to earth with an acceleration equal to g. Some of the ball s initial velocity will be directed in the x direction. Since there is nothing fighting the motion in the x direction (neglecting air resistance) the ball will have a constant velocity in the x direction. Being scientists, we need to know why a ball travels along a parabolic path. We also need a theory to be able to predict the trajectory of other balls, missiles, cows or any other projectile. To make this prediction, we need to look at two-dimensional kinematics. If you understand one-dimensional kinematics, it is not difficult to extend these ideas to motion in two dimensions. The idea is that: Two dimensional motion is two independent perpendicular, one-dimensional motions. Projectile Motion - 3

4 To illustrate what I mean, let s take a look at a very important type of motion called projectile motion. Projectile motion is the motion of an object under the influence of a uniform gravitational field, such as the field near the Earth s surface. Any object tossed into the air undergoes projectile motion (as long as you assume air resistance is negligible and can ignore the Earth s rotation). I just said that two-dimensional motion consists of two independent, perpendicular one-dimensional motions. So, let s examine the two perpendicular directions independently. First, consider the vertical direction. In the vertical direction, the Earth pulls the object downward under the influence of gravity. This causes the object to move with constant acceleration in the vertical direction. That is, the object s velocity increases downward in equal amounts in equal times. Examine the following diagram of a ball falling. The ball is shown at 1-second intervals: y 1 gt Figure 1: Constant acceleration in the vertical direction. As you can see, the ball moved a total of 1 unit of distance in the first second. After seconds, it has moved a total distance of 4 units, 9 total units after 3 seconds, and so forth. This is constant acceleration. Projectile Motion - 4

5 Next, consider the horizontal direction. In the horizontal direction, there is no pull of gravity. Therefore, the object s velocity does not change in the horizontal direction since nothing is pushing or pulling on it. This is called constant velocity motion. Here, the object moves equal distances in equal times. So, examine the diagram of a ball moving horizontally with constant velocity: x v Figure : Constant velocity in the horizontal direction. ox t The above diagram shouldn t be too surprising. The ball moves 1 unit every second and, therefore, there is equal spacing between each position of the ball shown. Now, projectile motion comes in when we combine the constant acceleration in the vertical direction with the constant velocity in the horizontal: x v ox t y 1 gt Resulting parabolic trajectory Figure 3: Constant velocity in the horizontal direction combined with constant acceleration in the vertical direction. We get a parabola! The parabola is the characteristic shape of uniformly accelerated motion, such as we have with a projectile. Projectile Motion - 5

6 So, if a ball is dropped from rest from a certain height (v ox, v oy, x o and y o = 0), it will hit the ground in a certain time, given by the expression: y 1 gt where y is the height, t is the time, and g is the acceleration due to gravity. So what happens to the time to hit the ground if we drop the ball from the same height, but give it some speed in the horizontal direction only (there is no vertical speed to start with)? The time to reach the ground is exactly the same! Why? Because the vertical and horizontal directions are independent. What goes on in one direction does not affect anything in a perpendicular direction. So now what happens if we launch a ball at some angle θ. Keep in mind that the vertical and horizontal directions are independent. The horizontal case remains the same as before. As for the vertical case, we have now introduced an initial vertical velocity. This just adds another factor to the vertical equation. 1 y gt v t (this is still assuming that y o =0) oy v 0 v oy θ v ox Now if we remove all assumptions we get the most general expressions for projectile motion: y Horizontal (y-dir.) Vertical (x-dir.) x v t x 1 gt voyt y o ox o Where, v ox = v cos and v oy = v sin o o Projectile Motion - 6

7 Equipment The ball moves too quickly to record with the human eye. Therefore, we will have to use some technology to capture its motion. We will be using a couple of new pieces of equipment in today s lab: two photogates hooked in series to the PASCO mini-launcher and a Time of Flight pad. Both will be explained below. Always load the launcher with the supplied plunger. NEVER point the launcher at anybody or place any part of your body in the path of a projectile! If you do not wear glasses you will need to wear goggles during this lab. Figure 4: Diagram of the important parts of the PASCO mini launcher. Projectile Motion - 7

8 PASCO mini launcher Time of Flight (TOF) pad. Figure 5: Experimental Setup diagram Logger Pro Software tidbits. The two photogates are connected in series and both entered into CH 1 of the LabPro box. Therefore in this example GateState 1 shows the triggering of both photogates. Status of photogates. Place your hand in the gates and notice how the status changes. This indicator is also available on the photogates, look for a red LED light. It will turn on when the gate is blocked. The Time of Flight pad is set up as the second photogate. It will send a Gate State of 1 when the ball hits the pad. In this example there are multiple 1 states, this happens when the ball bounces on the pad. You only need the time from the first impact. Using the distance between gates and this time interval you can calculate the initial velocity. This is the running count of the time since the Collect button was pressed. The time difference between these two events is the time of flight. Projectile Motion - 8

9 The Lab Part A: Setup Open the file: PES 116/Projectile Motion/Mini Launcher.cmbl. Position the photogates such that the first inferred beam will be broken as soon as the ball leaves the launcher. This should already be setup properly, but you never know if some other student has adjusted it. The photogates are mounted together on a bracket, which in turn is connected to the mini launcher with a thumbscrew on the bottom of the launcher. Loosen this thumbscrew and slide the whole bracket back until the light on the first photogate lights up (indicating that it is blocked) then move the bracket forward slightly until the light goes out and tighten the thumbscrew. Light is on back of photogates. thumbscrew The major problem with this lab and the part you need to pay the closest attention to is locating the origin. Mini-launcher set to proper angle and clamped down. Symbol on launcher indicating position of ball. Use a ruler to measure the location of y o. Plum bob locates x o. Mark the exact location on a piece of tape. Note: These steps will need to be repeated every time the angle is changed. Projectile Motion - 9

10 With the launcher setup it is time to test fire the launcher. If you are not wearing glasses then you must have on a pair of goggles. Place the steel ball in the magnetic cup inside the launcher. Use the plunger (plastic tube) to prime the launcher. We will be firing using Medium range ( Clicks in). Position the TOF (Time Of Flight) pad about meters down range with a catch box located behind the pad to capture the ball. The ball will get lost easily so having a lab partner near the landing to collect the ball will help. TOF pad Catch box Target packet locates exact location of landing. Fire the launcher with a quick pull on the yellow string. Note where the ball lands and readjust the location of the TOF pad such that the ball will hit it somewhere in the middle. At this point attach the Target packet (see the page below) to the pad in order to record the exact location of impact. This is also a good time to check that the software is reading all the sensors properly. So before the next firing click the button to start the data collection and monitor the computer readings. Fire the ball again and make further adjustments such that the ball will strike the impact tape. Note: it would be wise the x-out past impacts to avoid getting confused after multiple impacts. To find the separation between the photogates, use the gold arrows drawn on each of the photogates as the location of the light beam. Measure the distance between the arrows. You will need this measurement to calculate the initial velocity of the ball. Usually this is the end of the lab! I will however give you another lab write-up in worksheet form. Soon you will have to write your own. Projectile Motion - 10

11 Bull s eye Targets Note: Sandwich a carbon sheet between two of these bull s eye targets such that an impact will transfer a mark to the bottom target. Place the sandwich on top of the Time-of-flight pad. Circle and label the impact impression on the bottom sheet. Do not throw away the carbon sheet, other lab sections will need it. Projectile Motion - 11