1. 1. We know with initial , . Integrate with respect to time to get

      

      Initial conditions give . So which approaches the constant a/bm asymptotically in time:

      

      Another integration gives

      

      which setting equal to initially gives Note that this asymptotically approaches the line x = at/bm:

      

   2. We invert our expression above for velocity to obtain

      

      so that F = a - bmv .

      Interpreting this as the force on a ball dropping through viscous fluid gives a = -mg (constant gravitational force, taking our axis pointing upward and g to be positive). Part (a) gives us the terminal velocity

      

      so we interpret our physical force as

      

2. 1. For velocity-dependent force, we can use the integral equation

      

      Here so the integral gives

      

      which we invert to obtain

      

      Integrating and applying the initial condition x(0) = 0 gives

      Rearranging then exponentiating both sides gives , so we get

      

   2. We could rearrange Newton's second law to directly relate variables v and x, by exploiting the chain rule for assumed solution v(x):

      

      Thus Newton II

      

      becomes

      

      Both sides of this equation depend only on a single variable, so we can sensibly integrate them from the beginning of the motion to its current state:

      

      Exponentiating both sides we can solve for v(x):

      

      as in (b).

   3. Note that this result, comes to rest ( ) only in the infinite limit . From our expression from part (a) for x(t), only in the asymptotic limit, .

3. 1. We're given , and asked to find F(x). Since , we find it by differentiating, using the chain rule to convert x-derivatives to time derivatives:

      

      Substituting our original expression for back in gives F(x):

      

   2. To find x(t), we can rearrange our expressions for , grouping x-dependence and t-dependence on their own sides of the equation:

      

      Integrating both sides, and using our initial conditions to set integration constants,

      

      or, solving for x,

      

   3. To find F(t), we can just substitute our expression for x(t) into F(x):

      

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