Understanding Simple Kinematics: Solving for Velocity in Resisted Motion

[bcjochim07]

Homework Statement

I should know this, but it's been awhile since I've dealt w/ kinematics.

As the simplest example of resisted motion of a particle, find the velocity of horizontal motion in a medium in which the retarding force is proportional to velocity.

So Fr is something like -kmv, where k is a constant.

I'm tempted to use v=vo+at in this manner:

ma=-kmv, so a = -kv
then v=vo-kvt
v(1+kt)=vo
v=vo/(1+kt)

But my book uses integrals:
mdv/dt=-kmv
int(dv/v)=-k*int(dt)
lnv=-kt+C , v= c1e^-kt where (c1=e^C) and applying initial conditions you get
v=vo*e^-kt
and this makes a lot of sense to me.

So could somebody please refresh me on why I cannot solve for a and substitute into v=vo+at? I'm thinking it has to do with the constantly changing force, but I'm looking for a good explanation.

 

[rl.bhat]

Kinematic equation v = vo + at is applicable to a motion having constant acceleration. But in the given problem acceleration is not constant.

 

[tomcue]

Your approach using v=vo+at assumes a constant acceleration, which is not the case in this scenario. The retarding force is proportional to velocity, which means that as the velocity changes, so does the force and therefore the acceleration. This means that the acceleration is not constant and cannot be solved for using the equation a=v/t.

Instead, the correct approach is to use the equation F=ma, where F is the retarding force and m is the mass of the particle. In this case, the retarding force is -kmv, so we can rewrite the equation as ma=-kmv. From here, we can use the equation v=vo+at to solve for the velocity as a function of time.

Integrating both sides of the equation ma=-kmv with respect to time, we get m∫a dt=-km∫v dt. Since we know that a= dv/dt, this becomes m∫dv= -km∫v dt. Integrating both sides again, we get mv=-km∫v dt+C, where C is a constant of integration. This can be rearranged to get v=vo*e^-kt, where vo is the initial velocity at time t=0.

In summary, using the equation F=ma and integrating with respect to time allows us to solve for the velocity as a function of time in a scenario where the retarding force is proportional to velocity. This approach takes into account the changing force and acceleration, unlike the v=vo+at equation which assumes a constant acceleration.