Problem 4) Moving particle with friction driven by an impulse [50 points] The equation of motion for the velocity v(t) of a freely moving particle with friction and driven by an external impulse at time t=0 is given by dy dt + Av=J8(t) where is the friction constant. The particle is at rest for t<0 and is agitated by an externa impulse J8(t) at time t=0 where 8(t) is the Dirac delta function and J is a constant (the impulse might be caused by the collision with another particle at t=0). a) Use the Fourier transformations v(1)= — ] døg(w)e¹™ and 8(1) == doe¹ 27 to derive from (1) the following equation for g(): (iw+2)g(a) = J. Use this result to show: v(t) = do 2πi (1) v(t) = los Tưới @-12 [0 for <0 Je for t>0 b) Extend the integration contour in (2) to the complex - plane and calculate the integral using residues and Jordan's lemma to show: (2)
Problem 4) Moving particle with friction driven by an impulse [50 points] The equation of motion for the velocity v(t) of a freely moving particle with friction and driven by an external impulse at time t=0 is given by dy dt + Av=J8(t) where is the friction constant. The particle is at rest for t<0 and is agitated by an externa impulse J8(t) at time t=0 where 8(t) is the Dirac delta function and J is a constant (the impulse might be caused by the collision with another particle at t=0). a) Use the Fourier transformations v(1)= — ] døg(w)e¹™ and 8(1) == doe¹ 27 to derive from (1) the following equation for g(): (iw+2)g(a) = J. Use this result to show: v(t) = do 2πi (1) v(t) = los Tưới @-12 [0 for <0 Je for t>0 b) Extend the integration contour in (2) to the complex - plane and calculate the integral using residues and Jordan's lemma to show: (2)
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Solve both parts (a) and (b)
![Problem 4) Moving particle with friction driven by an impulse [50 points]
The equation of motion for the velocity v(t) of a freely moving particle with friction
and driven by an external impulse at time t=0 is given by
dv
dt
+ Av=J8(t)
where is the friction constant. The particle is at rest for t<0 and is agitated by an external
impulse J8(t) at time t=0 where 8(t) is the Dirac delta function and J is a constant
(the impulse might be caused by the collision with another particle at t = 0).
a) Use the Fourier transformations v(t)=— Ĵ døg(w)e'
to derive from (1) the following equation for g():
(iw+2)g (w) = J
Use this result to show: v(t) =
;
log(@)e'¹™ and 5(1) = —ƒ ª
2.
v(t) =
J
e love
žįdo S
da
2πi
@-12
(1)
(0 for <0
Je for 1>0
dwe
b) Extend the integration contour in (2) to the complex - plane and calculate the integral
using residues and Jordan's lemma to show:
(2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F121b5696-0956-4867-857e-f1ef2b6bfcb7%2F338d0573-8a61-4685-a456-cf9fa92f531f%2Fzv9atps_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 4) Moving particle with friction driven by an impulse [50 points]
The equation of motion for the velocity v(t) of a freely moving particle with friction
and driven by an external impulse at time t=0 is given by
dv
dt
+ Av=J8(t)
where is the friction constant. The particle is at rest for t<0 and is agitated by an external
impulse J8(t) at time t=0 where 8(t) is the Dirac delta function and J is a constant
(the impulse might be caused by the collision with another particle at t = 0).
a) Use the Fourier transformations v(t)=— Ĵ døg(w)e'
to derive from (1) the following equation for g():
(iw+2)g (w) = J
Use this result to show: v(t) =
;
log(@)e'¹™ and 5(1) = —ƒ ª
2.
v(t) =
J
e love
žįdo S
da
2πi
@-12
(1)
(0 for <0
Je for 1>0
dwe
b) Extend the integration contour in (2) to the complex - plane and calculate the integral
using residues and Jordan's lemma to show:
(2)
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