mathematical models\n61, hawking radiation according to the english theoretical physicist and cosmologist…
mathematical models\n61, hawking radiation according to the english theoretical physicist and cosmologist stephen hawking (1942 - 2018) the mass (m(t)) of a stellar mass called a black hole decreases over time due to radiation. a model for the stellar mass is given by (\frac{dm}{dt}=-\frac{k}{m^{2}}).\nsolve for (m(t)) when (m(0)=m_0).
Answer
Answer:
$m(t)=\left(m_{0}^{3}+3kt\right)^{\frac{1}{3}}$
Explanation:
Step1: Separate variables
$m^{2}dm=-kdt$
Step2: Integrate both sides
$\int m^{2}dm=-\int kdt$
Step3: Calculate integrals
$\frac{m^{3}}{3}=-kt + C$
Step4: Use initial - condition
When $t = 0$, $m=m_{0}$. Substituting into $\frac{m^{3}}{3}=-kt + C$, we get $\frac{m_{0}^{3}}{3}=C$.
Step5: Solve for $m(t)$
$\frac{m^{3}}{3}=-kt+\frac{m_{0}^{3}}{3}$, then $m^{3}=m_{0}^{3}+3kt$, so $m(t)=\left(m_{0}^{3}+3kt\right)^{\frac{1}{3}}$