Free Falling 1

Problem 1.

An object (its mass is $m$) falls through the air toward Earth. Assuming that the only forces acting on the object are gravity ($g$ is the gravity constant) and air resistance (proportional to the the speed of the object).

(a) Determine the speed $v(t)$ of the object.
(b) Show that $\displaystyle{\lim_{t\to\infty}}v(t)=v_{*}$ independently of the initial speed $v_0$, where $v_{*}$ is the equilibrium speed.

Electric Circuits 2

Fig.2. RLC-circuit.

 

Problem 2.

Determine $I(t)$ for
(a) a constant electromotive force,
(b) a periodic electromotive force,
if $I(0):=I_0$ and $I'(0)=I_0'$ are given.

Free Falling 2

Problem 2.

An object (its mass is $m$) falls through the air toward Earth. Assuming that the only forces acting on the object are gravity ($g$ is the gravity constant) and air resistance (proportional to the square of the speed of the object).

(a) Determine the speed $v(t)$ of the object.
(b) Show that $\displaystyle{\lim_{t\to\infty}}v(t)=v_{*}$ independently of the initial speed $v_0$, where $v_{*}$ is the equilibrium speed.

Electric Circuits 3

Problem 3.

Determine the particular solution $I_p(t)$ of the equation ($L,R,C,\gamma\gt 0$) \[ LI''(t)+RI'(t)+\frac{1}{C}I(t)=E_0e^{-\gamma t}. \]

Chemical Reactions 1

Problem 1.

The functions $x(t)$ and $y(t)$ describe the concentration of two substances, respectively. The first component transforms into the second one with the rate coefficient $k\gt 0$, and the second component decomposes at the rate $\mu\gt 0$. Describe the system.