An object with a mass m slides across a frictionless horizontal surface. At time

ID: 1778756 • Letter: A

Question

An object with a mass m slides across a frictionless horizontal surface.

At time t = 0 it has an initial velocity v = v0 and its initial position is x = 0.

A drag force Fd acts on the object and depends both linearly and quadratically on the velocity:

Fd = k1v k2v2 , where k1 and k2 are constant coefficients.

(a) Neglecting the gravitational force, find the instantaneous velocity, v(t), and position, x(t), of the object.

(b) Will the object eventually come to rest at a finite distance from where it enters? If so where does it stop?

a( retardation in this case) = F/m=- 1/m ( K1 V + K2 V^2)= dv/dt

dv = a dt

dv/ (K1 V + K2 V^2) = - 1/m dt

dv/ V ( K1 + K2 V) = -1/m dt

(1/K1 ) ln( V/(K1+K2V) + c = -t/m

at t=0, V = v0

-( 1/K1) ln ( V0/K1 + K2 V0) = c

( 1/K1) ln ( V / (K1+K2V) - (1/K1) ln ( V0/K1 + K2 V0)= -t/m

( 1/K1) ln ( V/ K1 + K2V) =- t/m + ( 1/K1 ) ln ( V0/K1 + K2 V0)------velocity

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