A particle of unit mass is projected vertically upwards in a medium whose resistance is \(k\) times the square of the velocity of the particle. If the initial velocity is \(u\), prove that the velocity \(v\) after rising through a distance \(s\) satisfies \begin{equation*} v^{2}=u^{2}\e^{-2ks}+\frac{g}{k}(\e^{-2ks}-1). \tag{*} \end{equation*} Find an expression for the maximum height of the particle above he point of projection. Does equation \((*)\) still hold on the downward path? Justify your answer.
Solution: \begin{align*} \text{N2}(\uparrow): && 1 \cdot v\frac{\d v}{\d s} &= -g - kv^2 \\ \Rightarrow && \int \frac{v}{g+kv^2} \d v &= \int -1 \d s \\ \Rightarrow && \frac{1}{2k}\ln(g+kv^2) &= -s + C \\ s =0, v = u: && \frac{1}{2k} \ln(g+ku^2) &= C \\ \Rightarrow && s &= \frac{1}{2k} \ln \frac{g+ku^2}{g+kv^2} \\ \Rightarrow && e^{-2ks} &= \frac{g+kv^2}{g+ku^2} \\ \Rightarrow && v^2 &= u^2e^{-2ks} + \frac{g}{k}(e^{-2ks}-1) \end{align*} The maximum height will be when \(v = 0\), ie \(\displaystyle s = \frac{1}{2k}\ln\left(1 + \frac{k}{g}u^2 \right)\). On the downward path the resistance will be going upwards, ie \begin{align*} \text{N2}(\uparrow): && 1 \cdot v\frac{\d v}{\d s} &= -g + kv^2 \end{align*} but our solution is solving a different differential equation, therefore unless \(k=0\) the equation will be different.
An experiment produces a random number \(T\) uniformly distributed on \([0,1]\). Let \(X\) be the larger root of the equation \[x^{2}+2x+T=0.\] What is the probability that \(X>-1/3\)? Find \(\mathbb{E}(X)\) and show that \(\mathrm{Var}(X)=1/18\). The experiment is repeated independently 800 times generating the larger roots \(X_{1}, X_{2}, \dots, X_{800}\). If \[Y=X_{1}+X_{2}+\dots+X_{800}.\] find an approximate value for \(K\) such that \[\mathrm{P}(Y\leqslant K)=0.08.\]
Solution: \((x+1)^2+T-1 = 0\) so the larger root is \(-1 + \sqrt{1-T}\) \begin{align*} && \mathbb{P}(X > -1/3) &= \mathbb{P}(-1 + \sqrt{1-T} > -1/3) \\ &&&= \mathbb{P}(\sqrt{1-T} > 2/3)\\ &&&= \mathbb{P}(1-T > 4/9)\\ &&&= \mathbb{P}\left (T < \frac59 \right) = \frac59 \end{align*} Similarly, for \(t \in [-1,0]\) \begin{align*} && \mathbb{P}(X \leq t) &= \mathbb{P}(-1 + \sqrt{1-T} \leq t) \\ &&&= \mathbb{P}(\sqrt{1-T} \leq t+1)\\ &&&= \mathbb{P}(1-T \leq (t+1)^2)\\ &&&= \mathbb{P}\left (T \geq 1-(t+1)^2\right) = (t+1)^2 \\ \Rightarrow && f_X(t) &= 2(t+1) \\ \Rightarrow && \E[X] &= \int_{-1}^0 x \cdot f_X(x) \d x \\ &&&= \int_{-1}^0 x2(x+1) \d x \\ &&&= \left [\frac23x^3+x^2 \right]_{-1}^0 \\ &&&= -\frac13 \\ && \E[X^2] &= \int_{-1}^0 x^2 \cdot f_X(x) \d x \\ &&&= \int_{-1}^0 2x^2(x+1) \d x \\ &&&= \left [ \frac12 x^4 + \frac23x^3\right]_{-1}^0 \\ &&&= \frac16 \\ \Rightarrow && \var[X] &= \E[X^2] - \left (\E[X] \right)^2 \\ &&&= \frac16 - \frac19 = \frac1{18} \end{align*} Notice that by the central limit theorem \(\frac{Y}{800} \approx N( -\tfrac13, \frac{1}{18 \cdot 800})\). Also notice that \(\Phi^{-1}(0.08) \approx -1.4 \approx -\sqrt{2}\) Therefore we are looking for roughly \(800 \cdot (-\frac13 -\frac{1}{\sqrt{18 \cdot 800}} \sqrt{2})) = -267-9 = -276\)
Mr Blond returns to his flat to find it in complete darkness. He knows that this means that one of four assassins Mr 1, Mr 2, Mr 3 or Mr 4 has set a trap for him. His trained instinct tells him that the probability that Mr \(i\) has set the trap is \(i/10\). His knowledge of their habits tells him that Mr \(i\) uses a deadly trained silent anaconda with probability \((i+1)/10\), a bomb with probability \(i/10\) and a vicious attack canary with probability \((9-2i)/10\) \([i=1,2,3,4]\). He now listens carefully and, hearing no singing, concludes correctly that no canary is involved. If he switches on the light and the trap is a bomb he has probability \(1/2\) of being killed but if the trap is an anaconda he has probability \(2/3\) of survival. If he does not switch on the light and the trap is a bomb he is certain to survive but, if the trap is an anaconda, he has a probability \(1/2\) of being killed. His professional pride means that he must enter the flat. Advise Mr Blond, giving reasons for your advice.
Solution: \begin{array}{c|c|c|c} & A & B & C \\ \hline 1 & \frac{1}{10} \cdot \frac{2}{10} & \frac{1}{10} \cdot \frac{1}{10} & \frac{1}{10} \cdot \frac{7}{10} \\ 2 & \frac{2}{10} \cdot \frac{3}{10} &\frac{2}{10} \cdot \frac{2}{10} &\frac{2}{10} \cdot \frac{5}{10} \\ 3 & \frac{3}{10} \cdot \frac{4}{10} &\frac{3}{10} \cdot \frac{3}{10} &\frac{3}{10} \cdot \frac{3}{10} \\ 4 & \frac{4}{10} \cdot \frac{5}{10} &\frac{4}{10} \cdot \frac{4}{10} &\frac{4}{10} \cdot \frac{1}{10} \\ \hline & \frac{2+6+12+20}{100} & \frac{1 + 4 + 9 + 16}{100} & \frac{7 + 10 + 9 + 4}{100} \end{array} Therefore \(\mathbb{P}(A) = \frac{4}{10}, \mathbb{P}(B) = \frac{3}{10}, \mathbb{P}(C) = \frac{3}{10}\), in particular, \begin{align*} \mathbb{P}(A | \text{not }C) &= \frac{4}{7} \\ \mathbb{P}(B | \text{not }C) &= \frac{3}{7} \\ \end{align*} If he switches the light on, his probability of survival is \(\frac47 \cdot \frac23 + \frac37 \cdot \frac12 = \frac{25}{42}\), if he doesn't his probability is \(\frac12 \cdot \frac47 +\frac37= \frac{5}{7} = \frac{30}{42}\) therefore he shouldn't switch the light on.
The maximum height \(X\) of flood water each year on a certain river is a random variable with density function \begin{equation*} {\mathrm f}(x)= \begin{cases} \exp(-x)&\text{if \(x\geqslant 0\),}\\ 0&\text{otherwise}. \end{cases} \end{equation*} It costs \(y\) megadollars each year to prepare for flood water of height \(y\) or less. If \(X\leqslant y\) no further costs are incurred but if \(X\geqslant y\) the cost of flood damage is \(r+s(X-y)\) megadollars where \(r,s>0\). The total cost \(T\) megadollars is thus given by \begin{equation*} T= \begin{cases} y&\text{if \(X\leqslant y\)},\\ y+r+s(X-y)&\text{if \(X>y\)}. \end{cases} \end{equation*} Show that we can minimise the expected total cost by taking \[y=\ln(r+s).\]