Question

An object of mass mis released from rest and falls under the influence of gravity. If the magnitude of the force due to air resistance is bvⁿ, where band nare 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 $\mathbf{\text{t}}\mathbf{\to }\mathbf{+}\mathbf{\infty }$

Short answer

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

Step-by-step solution

Find the limiting velocity of the object

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

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 role="math" localid="1663965910386" $\mathbf{v}\mathbf{=}\mathbf{(}\frac{\mathbf{mg}}{\mathbf{b}}{\mathbf{)}}^{\frac{\mathbf{1}}{\mathbf{n}}}$.