Physics Lab : Skydiving Mechanics, Euler’s Method In this lab, you will investigate the motion of a skydiver in free‐fall. I.
Without air resistance If we ignore air resistance and assume that the only force acting upon the skydiver is gravity, then we can write a very simple differential equation. •
State Newton’s second law and write this differential equation. •
Write the height of the skydiver as a function of time. If there were no air resistance, however, a parachute would be worthless and skydiving would be a once‐in‐a‐lifetime sport! II. With air resistance Air resistance produces an upward force that resists the downward motion. Introducing the air resistance into the model yields a very different differential equation: Where : C is the coefficient of air resistance d is the density of air A is the surface area of the body v is the instantaneous velocity m is the mass of the skydiver g is the acceleration due to gravity •
C=0,57 ‐3
d=1.3 kg.m 2
2 A= 0,7 m before the parachute is open, 40 m after m=75kg ‐2 g=9,81 m.s
Write the approximations made to get this differential equation. We assume that there is no relevant Archimede’s force throughout the motion. It is reasonable because the volume of the skydiver, with or without parachute is small enough and air density is much smaller than the skydiver’s density, even with equipment. Using this differential equation, you should be able to write the recursive formula used in Euler’s method to approximate the velocity. We will take our initial conditions to be t0=0 and v0=0. Copyright Emilangues We would also like to find values for the actual height of the skydiver at any time t. Using Euler’s method, since velocity is the rate of change of height with respect to time : We will assume the plane to be flying at 2000 m, hence h0=2000 m. •
Open a spreadsheet and use it to calculate acceleration, velocity and height step by step. From the initial jump to the time when the ripcord is pulled, the change in time will be 0.5 s. After the ripcord is pulled, the change in velocity is so dramatic that a much smaller Δt , Δt= 0,1 s is needed for Euler’s method to give a reasonable approximation of the skydiver’s behavior. A. First situation: without parachute let the skydiver just free‐fall for 50 seconds without opening his parachute at all. • Plot the graphs of velocity against time and height against time. Comment. • What is the terminal velocity reached ? • At what height is the skydiver after free‐falling for 35 seconds ? B. Second situation : with parachute Have the skydiver pull the ripcord after free‐falling for 30 seconds. Don’t forget to change the formulas and Δt after that time. • Plot the graphs of velocity against time and height against time. Comment. • What terminal velocity is reached after the skydiver opens his parachute? • How long did it take from when he opened his parachute until he hit the ground ? C. Third situation : with parachute, safer conditions In order for the skydiver to land safely, he must have his parchute open long enough to reach terminal velocity before impact with ground. A significantly higher velocity upon impact would cause physical injury. • You are to determine how long the skydiver can wait before pulling the ripcord and still land safely. • How far is he off the ground at that moment ? Prepare a written report which includes each of the following : Answers to all questions All requested print‐outs File name Copyright Emilangues A. Graph B. Graph Copyright Emilangues C. Graph Copyright Emilangues Table : With values With formulas ‐
Rows with closed parachute (situation 1) ‐
Rows with open parachute From row N°61(situation 2) From row N°74 (situation 3) Copyright Emilangues