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d²y dy dt² dt +c- Dirac delta function and F m- = -mg + Fod(t-1) in the 1. CUL

d²y dy dt² dt +c- Dirac delta function and F m- = -mg + Fod(t-1) in the 1. CUL

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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You do not need to show work for decomposing partial fractions
Ps) Consider a tennis ball falling in a vertical direction, subject to gravitational force and air resistance, described by the second-order differential equation (0.1) where m is the mass of the object, c is the friction coefficient, y(t) represents the height of the object above the reference point y = 0 which is the ground and g is the acceleration due to gravity. Additionally, the ball is released from rest at an initial height, that is y(0) = yo > 0, and with an initial velocity of zero, that is y' (0) = 0. (b) 401 para y (0) char no 0. m d²y dy + c dt² dt m = -mg for the uni Note that the limit as t goes to intim Now, a tennis player is going to hit (perfectly vertically) the tennis ball u ward at t = 1, incorporating an impulse in the ODE as follows dy d²y + c dt² dt 2011 = -mg + Fod(t-1) Ad on a well as the (0.2) where 8(t) is the Dirac delta function and Fo≥ 0 is the strength of the impulse. Use the Laplace transform to solve (0.2) with initial conditions y(0) = yo and y' (0) = 0. Your solution y(t) should depend on t as well as the parameters m, c, g, yo and Fo.
Transcribed Image Text:Ps) Consider a tennis ball falling in a vertical direction, subject to gravitational force and air resistance, described by the second-order differential equation (0.1) where m is the mass of the object, c is the friction coefficient, y(t) represents the height of the object above the reference point y = 0 which is the ground and g is the acceleration due to gravity. Additionally, the ball is released from rest at an initial height, that is y(0) = yo > 0, and with an initial velocity of zero, that is y' (0) = 0. (b) 401 para y (0) char no 0. m d²y dy + c dt² dt m = -mg for the uni Note that the limit as t goes to intim Now, a tennis player is going to hit (perfectly vertically) the tennis ball u ward at t = 1, incorporating an impulse in the ODE as follows dy d²y + c dt² dt 2011 = -mg + Fod(t-1) Ad on a well as the (0.2) where 8(t) is the Dirac delta function and Fo≥ 0 is the strength of the impulse. Use the Laplace transform to solve (0.2) with initial conditions y(0) = yo and y' (0) = 0. Your solution y(t) should depend on t as well as the parameters m, c, g, yo and Fo.
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