dv dt dt du b dt In( b +,/"e When t→>∞, the terminal speed of the object is v= 4. Assume the kinetic frictional force on a falling mass m is proportional to its speed v, with a proportionality constant B. Choose j to be vertically downward (a)Show that Newton's second law of motion yields dv,=mg-pvy (b)When the mass reaches its terminal sped, what is (c)Show that the terminal speed is v (d)If the mass is dropped from rest, show that v, (0=vem(l-e-pnm)D − mg = −ma dt dv ⇒ bv − mg = −m 2 dt m b dv b mg v = − − ⇒ 2 1 ∫ ∫ ∫ ∫ = − − ⇒ = − − ⇒ v t v t dt m b dv b mg v dt m b dv b mg v y 0 0 2 0 0 2 1 1 t m bg e b mg v b mg v t m b b mg v b mg v b mg 2 ln( ) 2 1 − = + − ⇒ = − + − ⇒ When t → ∞ , the terminal speed of the object is b mg v = 4. Assume the kinetic frictional force on a falling mass m is proportional to its speed v, with a proportionality constant β. Choose j ˆ to be vertically downward. (a) Show that Newton’s second law of motion yields y y mg v t v m = − β d d . (b)When the mass reaches its terminal sped, what is t vy d d ? (c) Show that the terminal speed is β mg vterm = . (d) If the mass is dropped from rest, show that ( ) (1 ) t / m y term v t v e−β = −