Question

Please help with Q1 a)b)c).

Question 1: In the lectures we considered simple projectile motion. Here we extend the description to include air resistance. For macroscopic objects in air, the dynamics equations including air resistance may be written V and ^- where m is the mass of the object, g is the acceleration due to gravity, y is the vertical direction, C is a dimensionless drag coefficient, A is the cross-sectional area of the object, pa 1.2kg/m3 is the density of air, and where V- v/, with2 Figure 1: In 1632, Galileo published the Dialogue on the Grea World Systems, which included discus- sion of projectile motion. Is the statement in the caption about the time of flight of cannon balls fired from a tower correct? Source: The Project 2Physics Course, Reader 1 Concepts of Motion (1970), by FJ Rutherford, G Holton, & FG. Watson.] FiG. 3.2. For cannon balls fired horizontally with different initial forward speeds, "all the balls in all the shots made horizontally remain in the air an equal time." (a) In 1632, Galileo argued that objects take the same time to fall a given distance, independent of their horizontal speed. Figure llustrates this claim. Based on Es.(I), s this true if air resistance is important? Briefly explain your logic. (b) Show analytically that a non-dimensional form of Eqs. (1) is dr dv where İGAL, is a non-dimensional drag parameter. Explain all steps in your working. (c) Is the Verlet method a suitable method to use to solve Eqs. (2)? If not, why not? Give a brief. but detailed, explanation. (d) Modify the codes from week 2 to solve the non-dimensional Eqs. (2) using the following method: To test your code, consider parameters appropriate for a ping pong ball: drag coefficient Ca 0.50, mass m2.7g, radius r 2.0cm, and solve the initial value problem from the first week: projection from ground level with initial speed 10 m/s and an initial angle 61- 60 degrees to the horizontal. Accurate integration of the equations should produce a range a bit less than half that for the case of simple projectile motion i.e. no air resistance) (i) What range do you obtain? (ii) Briefly explain how you chose your time step for an "accurate integration." iii Have your code estimate the time of flight, corresponding to the time at which the ping pong ball returns to ground level. What value do you obtain? Briefly explain how you made your estimate. (d) Modify your code from (c) to answer Galileo's cannon ball problem. Assume the parameters: drag coefficient Ca-0.10, mass m-3.0kg, radius r-4.5cm. Have your code solve the initial value problem for initial horizontal motion from a tower of height 20 m. (i) Run your code with initial speeds vi300 m/s and 600 m/s. Determine accurate values for the time of flight, and explain the basis for believing the results are accurate. (ii) Do your results confirm your expectations?

Answer 1

Answer:

given for macroscopic objects
   dr/dt = v v' [where v' is a unit vector along v]
   dv/dt = -0.5*Cd*rhoa*A*v^2 v' /m - g j [i, j and k are usual unit vectors]
   rhoa = 1.2 kg/m^3
   v = sqrt(vx^2 + vy^2)

a. along the x and the y axis if we segregate the given equations
  
   dx/dt = vx
   dy/dt = vy
   acceleration = a
   a = dv/dt = -0.5*Cd*rhoa*A*sqrt(vx^2 + vy^2){vxi + vyj}/m - gj

   as we can see that there is a horizontal component to acceleration, hence ax != 0
   also, ax depends on vx and vy, and ay depends on vx and vy as well
   since both vx and vy are dependent on ax and ay and ax and ay are dependent on vx and vy
   the initial speeds of the projectiles hence will change accelerations of the objects,
   this changes the vertical acceelration of fall and hence objects with different speeds shall take different time to reach the ground in presence of air resistance

b. consider a on dimensionalisinsg parameter Ls ( units of length)
   then
   r_ = r/Ls
   let time be non dimensionalised by T
   hence
   dr_/dt_ = (dr/dt)(T/Ls) = v_ = v/(Ls/T)
   hence velocity is normalised by Ls/T

   hence
   dv_/dt_ = (dv/dt)(T^2/Ls) = [-0.5*Cd*rhoa*A*v^2 v'/m - g j](T^2/Ls)
   now, let -0.5Cd*rhoa*A/m = D_/Ls
   then

   dv_/dt = [D_ * v^2 v' / Ls - g j ](T^2/Ls)
   dv_/dt = [D_ * v_^2 v' Ls^2 / Ls * T^2 - g j](T^2/Ls)
   dv_/dt = [D_ * v_^2 v' - Ls*g/T^2 j]
   now let
   Ls/T^2 = g
   T = sqrt(Ls/g)
   then

   dv_/dt = [D_ * v_^2 * v' - j]

c. Verlet integration is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics.

   to solve the above equations with verlet integration, we discretise the derivatives to find the value of velocity at a later period of time after integrating the derivative for velocity