PHYS5301, Aerodynamic Drag Forces In introductory mechanics we study projectile motion in an airless environment. The...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
PHYS5301, Aerodynamic Drag Forces In introductory mechanics we study projectile motion in an airless environment. The only forces we assume are from gravity. In this exercise we will study projectile motion in a realistic environment. The equations of motion depend on physical parameters that are determined empirically. The information needed is readily available online using NASA and engineering web sites. The exercise will ignore the change of gravitational acceleration for motion near the surface of the earth. The forces exerted on a mass, m. are W, the weight and the drag force Ď. F = Ď+W Ď+ W a= m In equation la the total force is, F. We want to determine the acceleration, a. Knowing the acceleration we follow the trajectory of m in steps of time, dt. This must be done numerically. Consider the displacement, r(t), and velocity, (t), at any given moment of time, t. du= adt (t + dt) = (t) + du dr = u(t)dt r(t + dt) r(t) + dr (la) (1b) (B) The drag equation (see for example a NASA website) is Ca(v)pv² A 2 You can employ the steps in equations 2a-2d in a computer code you write or use one of the mathematical packages available. By selecting a step size, dt you are deriving an analytical solution (by quadrature) according to the discus- sion in Finn's book, see equation 1.90, for example. You keep track of the total time. The physics is straightforward. In any given problem the issue is not so much a mathematical procedure but be aware of the physical condition of the particle at each step, dt. Equation 1.90 is for a simple case where the physical conditions of the particle are not time dependent. Following a 45 caliber bullet. Use MKS units! (A) A 45 caliber bullet has a mass of 13g and a cross sectional area of A = 1.06 x 10-4m². Assume the muzzle velocity is v= 275m/s. For this cal- culation ignore drag. What is the maximum height, in meters, and the time of flight to maximum height in seconds? At what time after firing does the bullet return to ground? Assume the initial velocity is vertical. D= (2a) (2b) (2c) (2d) (3) In equation 3 the coefficent of drag for this type of bullet depnds on the speed, u, through the air see figure 1, Ca(v) and on air density, p is the air den- sity. The direction of the drag force is opposite the velocity of the bullet. Note that if p = 0) we are back in the introductory physics situation. When you solve the problem, using whatever algorithm you chose, check, as a matter of princi- ple, that if p= 0 your algorithm gives you the answer in (A). Also, as a matter of practice, try a few steps dt to see if your answers agree within some criteria you choose for reliability of the procedure. In order to apply equation 3 you should consider the importance of the dependence of the air density, p(h), on al- titude, h, here's an example from figure 2 and on the local speed, u from figure 1. (Question1) What is the maximum height the bullet reaches and the time to maximum height if it starts from sea level? (Question2) What is the time it takes for the bullet to fall back to the ground and its speed? (Question3) Suppose the bullet is fired from Colorado Springs(altitude 1840m). What is the maximum height of the bullet and the time to maximum height? (Question4) What is the time it takes for the bullet to fall back to the ground and its speed in Colorado Springs? (Question5) What is the terminal velocity of the bullet? 18 values for G1 bullet v/mach1 drag coefficient 0. 0.266 0.244 0.242 0.222 0.218 0.21 0.11046 0.13676 0.28404 0.3156 0.37346 0.49444 0.61016 0.6838 0.70484 0.73114 0.7627 0.79952 0.8153 0.85738 0.8942 0.91524 0.95732 0.204 0.202 0.212 0.216 0.226 0.232 0.25 0.26 0.294 0.33 0.366 0.402 U Figure 1: Drag coefficient for a G1 bullet in air as a function of speed, v, mach where mach1 = 343m/s. - Altitude and Air Density Air density and specific volume as functions of altitude above sea level can be indicated as in the diagrams below: https://www.engineeringtoolbox.com/air-altitude-density-volume-d_195.html Density (kg/m3) 1.3 1.2 1.1 0.9 0.8 0 Density 500 Specific Volume engineeringtoolbox.com 1000 1500 2000 2500 Altitude above Sea Level (m) 1.3 1.2 1.1 0.9 0.8 3000 Specific Volume (m3/kg) Figure 2: Air density versus height above sea level from the engineeringtoolbox. PHYS5301, Aerodynamic Drag Forces In introductory mechanics we study projectile motion in an airless environment. The only forces we assume are from gravity. In this exercise we will study projectile motion in a realistic environment. The equations of motion depend on physical parameters that are determined empirically. The information needed is readily available online using NASA and engineering web sites. The exercise will ignore the change of gravitational acceleration for motion near the surface of the earth. The forces exerted on a mass, m. are W, the weight and the drag force Ď. F = Ď+W Ď+ W a= m In equation la the total force is, F. We want to determine the acceleration, a. Knowing the acceleration we follow the trajectory of m in steps of time, dt. This must be done numerically. Consider the displacement, r(t), and velocity, (t), at any given moment of time, t. du= adt (t + dt) = (t) + du dr = u(t)dt r(t + dt) r(t) + dr (la) (1b) (B) The drag equation (see for example a NASA website) is Ca(v)pv² A 2 You can employ the steps in equations 2a-2d in a computer code you write or use one of the mathematical packages available. By selecting a step size, dt you are deriving an analytical solution (by quadrature) according to the discus- sion in Finn's book, see equation 1.90, for example. You keep track of the total time. The physics is straightforward. In any given problem the issue is not so much a mathematical procedure but be aware of the physical condition of the particle at each step, dt. Equation 1.90 is for a simple case where the physical conditions of the particle are not time dependent. Following a 45 caliber bullet. Use MKS units! (A) A 45 caliber bullet has a mass of 13g and a cross sectional area of A = 1.06 x 10-4m². Assume the muzzle velocity is v= 275m/s. For this cal- culation ignore drag. What is the maximum height, in meters, and the time of flight to maximum height in seconds? At what time after firing does the bullet return to ground? Assume the initial velocity is vertical. D= (2a) (2b) (2c) (2d) (3) In equation 3 the coefficent of drag for this type of bullet depnds on the speed, u, through the air see figure 1, Ca(v) and on air density, p is the air den- sity. The direction of the drag force is opposite the velocity of the bullet. Note that if p = 0) we are back in the introductory physics situation. When you solve the problem, using whatever algorithm you chose, check, as a matter of princi- ple, that if p= 0 your algorithm gives you the answer in (A). Also, as a matter of practice, try a few steps dt to see if your answers agree within some criteria you choose for reliability of the procedure. In order to apply equation 3 you should consider the importance of the dependence of the air density, p(h), on al- titude, h, here's an example from figure 2 and on the local speed, u from figure 1. (Question1) What is the maximum height the bullet reaches and the time to maximum height if it starts from sea level? (Question2) What is the time it takes for the bullet to fall back to the ground and its speed? (Question3) Suppose the bullet is fired from Colorado Springs(altitude 1840m). What is the maximum height of the bullet and the time to maximum height? (Question4) What is the time it takes for the bullet to fall back to the ground and its speed in Colorado Springs? (Question5) What is the terminal velocity of the bullet? 18 values for G1 bullet v/mach1 drag coefficient 0. 0.266 0.244 0.242 0.222 0.218 0.21 0.11046 0.13676 0.28404 0.3156 0.37346 0.49444 0.61016 0.6838 0.70484 0.73114 0.7627 0.79952 0.8153 0.85738 0.8942 0.91524 0.95732 0.204 0.202 0.212 0.216 0.226 0.232 0.25 0.26 0.294 0.33 0.366 0.402 U Figure 1: Drag coefficient for a G1 bullet in air as a function of speed, v, mach where mach1 = 343m/s. - Altitude and Air Density Air density and specific volume as functions of altitude above sea level can be indicated as in the diagrams below: https://www.engineeringtoolbox.com/air-altitude-density-volume-d_195.html Density (kg/m3) 1.3 1.2 1.1 0.9 0.8 0 Density 500 Specific Volume engineeringtoolbox.com 1000 1500 2000 2500 Altitude above Sea Level (m) 1.3 1.2 1.1 0.9 0.8 3000 Specific Volume (m3/kg) Figure 2: Air density versus height above sea level from the engineeringtoolbox.
Expert Answer:
Answer rating: 100% (QA)
This is a multipart question regarding the aerodynamic drag forces on a 45 caliber bullet Ill break down the problem into smaller parts to provide a c... View the full answer
Related Book For
Statistics Unlocking The Power Of Data
ISBN: 9780470601877
1st Edition
Authors: Robin H. Lock, Patti Frazer Lock, Kari Lock Morgan, Eric F. Lock, Dennis F. Lock
Posted Date:
Students also viewed these mathematics questions
-
Planning is one of the most important management functions in any business. A front office managers first step in planning should involve determine the departments goals. Planning also includes...
-
The Crazy Eddie fraud may appear smaller and gentler than the massive billion-dollar frauds exposed in recent times, such as Bernie Madoffs Ponzi scheme, frauds in the subprime mortgage market, the...
-
The following additional information is available for the Dr. Ivan and Irene Incisor family from Chapters 1-5. Ivan's grandfather died and left a portfolio of municipal bonds. In 2012, they pay Ivan...
-
Ask survey subjects to estimate the length of 1 minute without looking at a watch or clock. Each subject should say go at the beginning of the minute and then stop when he or she thinks that 1 minute...
-
The account balances for Allied Electrical Supply, Inc., for the year ended August 31, 2012, are presented next in random order: Requirements 1. Prepare Allied Electrical Supplys single-step income...
-
Verify the relationship between \(M\) and \(M^{*}\), i.e., \(M M^{*}=M^{*}\), given below (4.7). Hint: use the fact that \(Z=Z^{*} I^{*}\) with \(I^{*}=\left(\iota_{N} \otimes I_{K^{\prime}}ight)\)....
-
Show the relationship between the work breakdown structure (WBS) and project costs.
-
A straddle occurs when an investor purchases both a call option and a put option. Such a strategy makes sense when the individual expects a major price movement but is uncertain as to the direction....
-
You are asked to review amounts paid to employees at a state agency. You do not have sufficient time to review all the employees listed below in detail; but you do have time to look at supporting...
-
The chief financial officer for Eagles Beach Wear and Gift Shop is planning for the companys cash flows for the next six months. The following table summarizes the expected accounts receivables and...
-
Swiss franc Option Prices Quoted as U.S. cents per Swiss franc, and each option contract consists of 62,500 Swiss francs. Calls-Last Puts-Last Option & Strike Underlying Price 58.51 58.51 58.51 58...
-
Why is diversity important in the composition of a governing board?
-
What are some budget types used in healthcare organizations?
-
Consider the following statement: NPV and IRR are expected values. Do you agree? Why or why not?
-
What are the three major categories of fraud?
-
What strategies do companies use to limit the cost of zero-based budgeting?
-
Rainy Day Corp. produces and distributes umbrellas around the world to both wholesalers and direct retailers. Rainy Day Corp. sells most of the products on credit to its customers. The December 31,...
-
Discuss whether responsible human resources management should apply different standards for the home company and suppliers, for developed countries and developing countries, and for large companies...
-
The Pew Research Center conducted a survey of randomly sampled American adults in 2008 and in 2010, asking them about their use of social networking sites such as Facebook. Table 7.31 shows age...
-
Match the scatterplots in Figure 2.55 with the correlation values. Figure 2.55 r = 0.89 Scatter Plot Scatter Plot 4 Scatter Plot + Scatter Plot -1 -1 -1 -1 -2 -2 -2 -2 -2 -1 -2 -1 -2 -1 -1 -2 (a)...
-
Complete each tree diagram by filling in the missing entries (marked with a ?). Case A Case B 0.225 Case C Case I 0.16 0.45 Case II Case A Case B 0.025 Case C 0.025
-
The diathermal wall (a) Is incapable of exchanging heat with the surroundings (b) Permits the full flow of heat from the system to the surroundings and vice versa (c) Both (a) and (b) (d) None of...
-
What is the effect of pressure on equilibrium conversion of a gas-phase chemical reaction?
-
The total energy of a system comprises (a) Kinetic energy, potential energy and vibrational energy (b) Kinetic energy, potential energy and rotational energy (c) Kinetic energy, potential energy and...
Study smarter with the SolutionInn App