Solve the problem.
-The kinetic energy $K$ of an object with mass $m$ and velocity $v$ is $K = \frac { 1 } { 2 } m v ^ { 2 }$. How is $\mathrm { dm } / \mathrm { dt }$ related to $\mathrm { dv } / \mathrm { dt }$ if $K$ is constant?
A) $\frac { d v } { d t } = - \frac { 2 m } { v } \frac { d m } { d t }$
B) $\frac { d m } { d t } = \frac { m } { v } \frac { d v } { d t }$
C) $\frac { d m } { d t } = - 2 m v ^ { 3 } \frac { d v } { d t }$
D) $\frac { \mathrm { dm } } { \mathrm { dt } } = - \frac { 2 \mathrm {~m} } { \mathrm { v } } \frac { \mathrm { dv } } { \mathrm { dt } }$

Question

Quizplus · Accepted Answer

Taking the derivative of the kinetic energy equation with respect to time:
\begin{align*} \frac { dK } { dt } & = \frac { d } { dt } \left( \frac { 1 } { 2 } m v ^ { 2 } ight) \ \frac { dK } { dt } & = \frac { 1 } { 2 } \left( 2mv \frac { dv } { dt } ight) \ \frac { dK } { dt } & = m v \frac { dv } { dt } \end{align*}
Since kinetic energy is constant, its derivative with respect to time is zero:
\begin{align*} \frac { dK } { dt } & = 0 \ m v \frac { dv } { dt } & = 0 \ \frac { dm } { dt } v ^ { 2 } & = -2mv \frac { dv } { dt } \ \frac { dm } { dt } & = -2m \frac { v } { 2v } \frac { dv } { dt } \ \frac { dm } { dt } & = -2 \frac { m } { v } \frac { dv } { dt } \end{align*}
Thus, the answer is D.

Solve the Problem KKK Of an Object with Mass mmm And Velocity vvv

Solve the Problem $K$ Of an Object with Mass $m$ And Velocity $v$