HOWEVER, PLEASE BASED ON ACTIVITY A TO COMPLETE ACTIVITY B
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Run Number Maximum height reached, y (cm) Net vertical displacement h-y-y (cm) Deviation Ah-h₁- hav (cm) Deviation squared (Ah)² (cm²) 1 224.4 2 225.2 3 230.1 4 228.3
5 226.3 6 228.1 7 229.9 8 224.1 9 227.3
9 227.2 10 226.3 11 225.1 12 229.1 Mean
Put in your units 2. Calculate the standard deviation & in the space below. Show units with your numbers. 4. You are going to use the kinematic equation vy vy²-2g(y-y) to find the initial speed of the ball. Note that vy=0 when the projectile reaches its maximum height. a. Show on the figure to the right the path of the projectile and coordinate axes that you will use. The origin position is important. Please do your work in pencil. b. Use the mean net vertical displacement (y-y0) of the ball and apply the equation to find the initial vertical speed (v0y) of the ball (when it becomes a projectile-leaves the gun). Show all of your work in the space below: equation used, algebra, numbers and units. Please do your work in pencil. 5. Calculate the percent difference between the launch speed calculated from the horizontal firing procedure (step 4 of part A) and the launch speed calculated from the vertical firing procedure (step 4 of part B) Percent Difference = (v₁ - V.)/(V)*100% Vav-(Vn+V.)/2
Projectile Motion Purpose: To analyze the motion of a projectile using the one-dimensional equations of kinematics, and to look at the statistics associated with random errors Materials: Projectile gun, one-meter stick, a two-meter stick, white paper, carbon paper, masking tape. Introduction: If air resistance is neglected, then projectile motion consists of simultaneous motions: horizontal motion with constant velocity and vertical motion with constant acceleration. For each direction the kinematics equations apply: x= x +Vt [1] y=y+vyt-(1/2)gt² [2] vy²=vy²-2g(y-y,) [3] Systematic errors Systematic errors are errors that are the result of a bias either in the system being measured or in the way that it is measured. A systematic error tends to shift the measured result in one direction - tending to be either above or below the true value of the quantity being measured. Lengths measured with a meter stick that has 0.2 cm worn off of the end have a systematic error. Random errors Random errors are those produced by unknown and unpredictable variations in an experimental situation. They can be variations in the behavior of the subject of the experiment or due to variations in the use of the measuring instruments (not mistakes in use, but unavoidable variations). In spite of a variety of causes, all random errors have one thing in common; they are as likely to produce a high result as a low result, which means they are truly random. If set of repeated measurements of the same parameter is made, such as the range of a gun fired under identical conditions each time, then the average of all of the values should give a better estimate than any one individual measurement. Random errors obey the laws of probability. This means that the as the number of measurements increases, the likelihood increases that the mean (average) value equals the true value. So the mean of 100 readings is more trustworthy than the mean of 10 readings. One reading alone is very unreliable.
12 N 13 14 15 16 17 18 19 20 21 22 23 24 25 We account for random errors by repeating our measurements, and taking an average over several measurements. If we make a set of measurements (x1, x2, x3,...), where N is the number of measurements, the mean value is given by: 1-N X=1=1 N The deviation, Ad, = x₁ - x, describes the variation of individual measurements from the mean value. If the xi value is above x then Adi is positive. If the xi value is below x then Adi is negative. The mean deviation is always zero because there are as many values above the mean as below. One way of getting an idea of the average deviation magnitude is to find the mean of the squares of the individual deviations and then take the square root of the mean of the squares. This is called the standard deviation, G. o=√(Ad) ave Activity A-HORIZONTAL FIRING PROCEDURE General Procedure - In this section you will use the SAME gun that you will use for the vertical firing later on in part B. This time you will fire the gun horizontally. You will use the average horizontal distance traveled by the ball to calculate the initial speed of the ball. Detailed Procedure 1. Clamp the gun base to the lab table. 2. Put the ball in the gun rod (barrel) and push it in with the loading baton until you hear a click to cock the gun. 3. Perform several trial firings and note the region on the floor where the ball lands. Tape a white sheet of paper to the floor over the region where the ball lands. Then place a sheet of carbon paper on top of the white paper. The ball will leave a mark on the white sheet each time it lands. 4. Fire the ball, lift the top sheet and number the mark as #1. Repeat and number as #2. Repeat until you have performed 10 runs. 5. Look at the physical setup and at the figure on page A-3 then answer all of the following questions for the horizontal firing. a) What is the vertical component of the initial velocity of the ball (at the instant it leaves the gun) when it is fired horizontally? Put in a numerical value. 0 m/s
b) What is the vertical component of the acceleration of the ball after it leaves the gun when it is fired horizontally? Neglect air resistance. Put in a numerical value. -9.8 m/s² c) What is the horizontal component of the acceleration of the ball after it leaves the gun when it is fired horizontally? Neglect air resistance. Put in a numerical value. 0 m/s² d) When determining the distance the ball traveled, it should be measured from the same point on the ball in the initial and final position. What part of the ball marks the paper on the floor: front, bottom, or back? front portion e) From what point on the ball should your measurement start when the ball is loaded in the gun: front, bottom, or back? Front f) When measuring the initial position of the ball, should the gun be in the cocked or un-cocked position (where does the ball become a projectile)? uncocked. The ball becomes a projectile when it leaves the gun. The ball leaves the gun when the gun becomes uncocked from a cocked position. Therefore, we need to measure from the uncocked position. 6. In the next step, you will be recording data on a data sheet. Do no calculations before recording the numbers. Use the table on the next page as your data sheet. Read and record all distances to the closest 0.1cm. 7. Measure and record on the next page the height of the ball above the floor when it is loaded in the gun to get the initial height of the ball. 8. Measure the horizontal flight distance in cm of the ball for each run. 9. Once you have all of your horizontal flight distances, remove the white sheet from the floor and return the carbon paper. HORIZONTAL FIRING ANALYSIS - Each person must do all of the following steps. Work independently and compare numbers to eliminate mistakes. 1. Complete the table below by doing the following: a. Calculate the mean (average) distance for the 10 firings. b. Calculate the deviation of each run from the mean. Note that the deviation is the distance for that run minus the mean distance. Deviation can be positive or negative. c. Calculate the mean (average) of the deviations. d. Ideally, what should the mean of the deviations equal? Put your answer here e. Square each deviation. f. Calculate the mean of the squared deviations. 227.97
43 44 45 40 47 48 49 828822 # Deviati Squared (Ad (cm) 52 53 Run Number ST 4. Horizontal distance (cm) Deviation Add d. (cm) e Square each deviation. f. Calculate the mean of the squared deviations DATA AND RANGE CALCULATIONS, HORIZONTAL FIRING Gan ID Number Initial height of the ball (yok 1 221.5 6:47 2 123.5 cm 231.7 3 234.5 6.53 4 2278 0.17 5 225.1 2.87 41.8609 13.9129 42.6400 0.0289 2360 2) Calculate the standard deviatione in the space below. Show units with your numbers. This is a measure of the scatter of your data values. The larger the variation in your values, the larger the value of (41.36 13.91 +42.64 0.0289 82369 12.4609 0.8469 23.7169 +12.0409+9.7969)/ 10 6 231.5 -3.53 12.4609 7 228.9 -0.93 0.8649 . 223.1 487 23.7169 9 2245 12.0400 10 2311 -3.13 9,7969 Mean 227.97
You are going to apply the kinematics equations [1] and [2] to find the initial horizontal speed of the ball. a) Show on the figure below, the path of the projectile from the gun to the floor and coordinate axes that you will use. The origin position is important. Please do your work in pencil. impact paper backstop b) Apply the equations using the ball's initial height and mean horizontal distance to find the initial speed of the ball (when it becomes a projectile - leaves the gun). Show all of your work in the space below: equations used, algebra, numbers and units. Do the calculation once using the mean (average) horizontal distance. Please do your work in pencil. Let the initial velocity = u We have t = √h/2g =>1=√1/(2x9.8) => t 0.2258 sec. Distance travel = vt = v(0.2258) Distance from the table = 227.97 cm = 2.22797 m =>v(0.2258) = 2.2797=> v- 10.0961 m/s V = 10.0916 m/s