Q3-3.4-14E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

An object of mass m is released from rest and falls under the influence of gravity. If the magnitude of the force due to air resistance is bvⁿ, where b and n are positive constants, find the limiting velocity of the object (assuming this limit exists). [Hint: Argue that the existence of a (finite) limiting velocity implies that  $\frac{\mathbf{dv}}{\mathbf{dt}}\mathbf{\to }\mathbf{0}$ as  $\mathbf{\text{t}}\mathbf{\to }\mathbf{+}\mathbf{\infty }$

Step-by-Step Solution

Answer

The limiting velocity is $\mathrm{v}=(\frac{\mathrm{mg}}{\mathrm{b}}{)}^{\frac{1}{\mathrm{n}}}$ .

Step 1: Find the limiting velocity of the object

According to the newton’s second law of equation  $\mathrm{m}\frac{\mathrm{dv}}{\mathrm{dt}}=\mathrm{mg}-{\mathrm{bv}}^{\mathrm{n}}$

Step 2: Apply given conditions

Let $\frac{\mathrm{dv}}{\mathrm{dt}}=0$  and   $\text{t}\to +\infty$

 $$\begin{gathered} 0=\mathrm{mg}-{\mathrm{bv}}^{\mathrm{n}} \\ \mathrm{v}=(\frac{\mathrm{mg}}{\mathrm{b}}{)}^{\frac{1}{\mathrm{n}}} \end{gathered}$$

 

Therefore, the limiting velocity is   $\mathbf{v}\mathbf{=}\mathbf{(}\frac{\mathbf{mg}}{\mathbf{b}}{\mathbf{)}}^{\frac{\mathbf{1}}{\mathbf{n}}}$.