Lap2009c&p104-ode2.5 ht
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This document contains a physics problem about modeling the trajectory of a home run hit by Willie Stargell at Dodger Stadium in 1969. It provides initial conditions and differential equations to model the motion, and asks the student to use calculus functions on a graphing calculator to solve for various values like the time and position when the ball hits the ground or wall. The teacher's notes indicate this problem is introducing the use of deSolve() to solve differential equations modeling exponential growth/decay and Newton's law of heating and cooling.
Lap2009c&p104-ode2.5 ht
- 1. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction Reconsider the Baseball problem and forget about that major cataclysm. You must have had a nightmare! Maybe it was that burrito you had at Taco Bell? Given Initial Conditions: (0) = (0) = (0) = ft Given Differential Model: (t) = These initial conditions and differential equations are supposed to model the first homerun ever hit at Dodger Stadium. You will test this model to see if it predicts Willie Stargell’s homerun in 1969! Remember that: (t) = = = You will be calculating: (t) = = = You will be calculating and analyzing: (t) = = ft a:C&P104.5Ht
- 2. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (1) Use deSolve() to find the General Solution x ’(t). Apply the given Initial Conditions to find the Particular Solution x ’(t). a:C&P104.5Ht
- 3. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (2) Use deSolve() to find the General Solution x (t). Apply the given Initial Conditions to find the Particular Solution x (t). a:C&P104.5Ht
- 4. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (3) Use deSolve() to find the General Solution y ’(t). Apply the given Initial Conditions to find the Particular Solution y ’(t). a:C&P104.5Ht
- 5. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (4) Use deSolve() to find the General Solution y (t). Apply the given Initial Conditions to find the Particular Solution y (t). a:C&P104.5Ht
- 6. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (5) Use Solve() and FIX9 to estimate tr>0 such that y(tr)=0. This is the time it takes for the ball to hit the ground. Use your estimate for tr to estimate x(tr). This is the horizontal range of flight. Willie’s homerun had a horizontal range of 506.5ft. This model should predict this value much more accurately than the UAM model of C&P103. a:C&P104.5Ht
- 7. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (6) The outfield wall at Dodger Stadium is 8ft high and is 395ft from the batter. Use Solve() and FIX9 to estimate tw such that x(tw) = 395. This is the time it takes for the ball to go over the wall. Use your estimate for tw to estimate y(tw). Did Willie Stargell get a homerun based on this model? a:C&P104.5Ht
- 8. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (7) What is the maximum height, or vertical range, for this trajectory? Use Solve() and FIX9 to estimate tm such that y ’(tm) = 0. Use your estimate for tm to estimate x(tm) and y(tm), the point of maximum height. a:C&P104.5Ht
- 9. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (8) Use Parametric Mode to graph the ballistic trajectory: (t) = . You should have a complete graph of the ball’s path, from when its hit by the bat to when it hits the ground using the following window: tmin = 0, tmax = 7, xstep = .1 xmin = 0, xmax = 600, xscl = 100 ymin = 0, ymax = 200, yscl = 100 Label the points when t=0, t=tr, t=tw and t=tm. Is this path parabolic? a:C&P104.5Ht
- 10. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (9) Estimate (tr) and |(tr)|. What angle does the ball’s trajectory make with the ground at the point of impact? Interpret these results. a:C&P104.5Ht
- 11. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction (10) Estimate and interpret this result. a:C&P104.5Ht
- 12. Calculus & Physics 104 Name: Exponential Decay and Exponential Approach nonUniformly Accelerated Motion: Free Fall with Friction Teacher’s notes: Introduce the TI89 function deSolve() to investigate the general solutions to differential equations modeling: Exponential Growth and Decay Newton’s Law of Heating and Cooling (Exponential Approach) Logistic Growth and Decay Do word problems from exercises in section 11.5 of Hughes-Hallett’s Calculus: Single Variable 4th ed. © 2005 from Wiley. a:C&P104.5Ht