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)