Question

A body of mass $m={10}^{2}Kg$ is moving in a medium and experience a frictional force $F=-K{v}^2$ . Its initial speed is $v_o=10m{s}^{-}1$ If, after 10 s, its energy is $\frac{1}{8}m{v}_o^2$, the value of K will be:

• A. ${10}^{-}Kg{s}^{-}1$
• B. ${10}^{-}4Kg{m}^{-}1$
• C. ${10}^{-}1Kg{m}^{-}1s^1$
• D. ${10}^{3}Kg{m}^{-}1$

Answer 1

The correct option is B ${10}^{-}4Kg{m}^{-}1$

Given that,

Mass $m={10}^{2}kg$

Initial velocity $v_0=10m/s$

Time $t=10s$

Frictional force $F=-K{v}^2$

Now,

$\frac{1}{2}m{v}_f^2=\frac{1}{8}m{v}_0^2$

$v_f=\frac{v_0}{2}$

$v_f=\frac{10}{2}=5m/s$

Now, the force is

$\left({10}^{-2}\right)\frac{dv}{dt}=-k{v}^2$

$\underset{10}{\overset{5}{\int }}\frac{dv}{v^2}=-100k\underset{0}{\overset{10}{\int }}dt$

$\frac{1}{5}-\frac{1}{10}=100k\times 10$

$k={10}^{-4}$

Hence, the value of k is ${10}^{-4}$