Raindrop Problem

QUESTION:

A spherical raindrop falling under constant gravity, grows by absorption of moisture from it's surroundings. The rate of increase of it's mass is proportional to the instantaneous surface area. Assuming that the raindrop starts with an infinitely small radius, determine its acceleration at time t.

SOLUTION:

\frac{dm}{dt}=4\pi r^{2}=\lambda r^{2}

where, m=\frac{4}{3}\pi r^{3}\rho

\frac{d}{dt}\left(\frac{4}{3}\pi r^{3}\rho\right)=\lambda r^{2}

\Rightarrow\frac{4}{3}\pi\rho3r^{2}\frac{dr}{dt}=\lambda r^{2}

\Rightarrow\frac{dr}{dt}=\frac{\lambda}{4\rho\pi}=\mu

\therefore r=\mu t+c

\therefore t=0 r=0

c=0

Therefore, r=\mu t

\frac{dm}{dt}=\lambda r^{2}

\Rightarrow\frac{1}{m}\frac{dm}{dt}=\frac{\lambda}{m}r^{2}

\Rightarrow\frac{1}{m}\frac{dm}{dt}=\frac{\lambda r^{2}}{\frac{4}{3}\pi r^{3}\rho}

=\frac{3\lambda}{4\pi\rho\left(\mu t\right)}=\frac{3\mu}{\mu t}=\frac{3}{t}

\Rightarrow\frac{d}{dt}\left(mv\right)=mg

\Rightarrow m\dot{v}+v\dot{m}=mg

\dot{v}+\frac{\dot{m}}{m}v=g

\frac{dv}{dt}+\frac{\dot{m}}{m}v=g

\Rightarrow\frac{dv}{dt}+\frac{3}{t}v=g

t\frac{d^{2}x}{dt^{2}}+3\frac{dx}{dt}-gt=0

let us consider the ansatz, x=At^{n}

\frac{dx}{dt}=Ant^{n-1}

\frac{d^{2}x}{dt^{2}}=An\left(n-1\right)t^{n-2}

So An(n-1)t^{n-1}+3Ant^{n-1}-gt=0

Therefore, n=2

A=\frac{g}{8}

x=\frac{g}{8}t^{2}

\frac{d^{2}x}{dt^{2}}=\frac{g}{4}

 

QUESTION:

Many chemical reactions have an activation energy E_{act}  which is about \frac{1}{2}eV . Hence Show that rate of chemical reactions doubles when temperature is increased by 10^{o} from the room temperature.

SOLUTION:

At T=300K,  which is about room temperature, the probalility that a particular reaction occurs is proportional to \exp\left(-\frac{E_{act}}{K_{B}T}\right).
If the temperature is increased to T+\Delta T=310K,

The probability increases to

\exp\left(-\frac{E_{act}}{K_{B}\left(T+\Delta T\right)}\right)

=\frac{\exp\left(-\frac{E_{act}}{K_{B}\left(T+\Delta T\right)}\right)}{\exp\left(-\frac{E_{act}}{K_{B}T}\right)}

=\exp\left(\frac{E_{act}\Delta T}{K_{B}T}\right) \approx2

QUESTION:

  Show that the error in truncating a convergent alternating infinite series after n terms is less than the ( n+1 ) th terms.

[Calcutta University Physics Model Solution]

SOLUTION:

calcutta university physics model question paper solved