Sketch the Following Functions

高中物理教学

1. $f[v] = 4 \pi v^2 \left(\frac{m}{2\pi k T}\right)^{3/2}{\rm e}^{-\frac{mv^2}{2 k T}}$, calculate $\int_{0}^{\infty} vf[v] {\rm d} v$, explain: why this is the mean speed of the particles? (Hint: suppose you have $n_i$ particles at speed $v_i$, show that the mean speed of the particles is: $\frac{\sum_i n_i v_i}{\sum_i n_i}$.)
2. $v[t] = v_{\rm f} + (v_{\rm i}-v_{\rm f}){\rm e}^{-t/t_0}$, calculate and sketch $x[t]$. What is the physical picture behind this $v[t]$ function?
3. $(E[p])^2=p^2c^2 + m^2c^4$, calculate $\displaystyle{\lim_{p \to \infty}}\frac{{\rm d}E}{{\rm d} p}$ (Hint: this is the relativistic energy-momentum relationship.) Show that for small $p$, $E[p]-mc^2 \approx \frac{p^2}{2m}$. Is this consistent with the Newton's energy-velocity relationship $E=\frac{1}{2}mv^2$? Why?
4. $A[\omega]=\frac{f}{\sqrt{(\omega_0^2-\omega^2)^2+2\beta^2\omega^2}}$. Find the maximum of $A$. Suppose $x[t] = A {\rm Cos}[\omega t + \phi]$, where ${\phi = \rm ArcTan}[\frac{-2\beta\omega}{\omega_0^2-\omega^2}]$, what is the resonance of the displacement? Calculate and sketch the relationship between the amplitude of velocity $v = \frac{{\rm d} x}{{\rm d} t}$ and $\omega$, what is the resonance of the velocity? Sketch the relationship between $\phi$ and $\omega$.
5. Sketch the gravitational potential $U[r]$ of a uniform sphere with radius $r_0$ and density $\rho$, suppose $\displaystyle{\lim_{r \to \infty}} U[r] = 0$. Hint: you should do the integral of $F[r]$.
6. Suppose you are having 2 charges $+q$ and $-q$ and $+x_0$ and $-x_0$, calculate the potential $U[x]$ at any $x$. Sketch the potential. (Hint: this model is called the electro-dipole)