The horizontal motion of the plunger and shall is arrested b

The horizontal motion of the plunger and shall is arrested by the resistance of the attached disk which moves through the oil bath. If the velocity of the plunger is upsilon_0 in the position A where x = 0 and t = 0, and if the deceleration is proportional to upsilon so that a = -k upsilon, derive expressions for the velocity upsilon and position coordinate x in terms of the time t. Also express upsilon in terms of x.

Solution

a = -kv
dv/dt = -kv
dv/v = -k dt
integrate on both sides
lnv(t) = -kt + lnC
v = v_o when t = 0
C = v_o
v(t) = v_oe^-kt

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dx/dt = v_oe^-kt
dx =v_oe^-kt dt
integrate on both sides.
x(t) = (-v_o/k)e^-kt + C\'
use x(0) = 0 = -v_o/k + C\' so C\' = v_o/k
x(t) = v_o(1 - e^-kt)/k

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e^-kt = 1 - kx(t)/v_o
v(t) = v_o - kx(t)