“The wave” is a particular type of pulse that can propagate through a large crowd gathered at a sports arena. The elements of the medium are the spectators, with zero position corresponding to their being seated and maximum position corresponding to their standing and raising their arms. When a large fraction of the spectators participates in the wave motion, a somewhat stable pulse shape can develop. The wave speed depends on people’s reaction time, which is typically on the order of 0.1 s. Estimate the order of magnitude, in minutes, of the time interval required for such a pulse to make one circuit around a large sports stadium. State the quantities you measure or estimate and their values.

Older auto-focus cameras sent out a pulse of sound and measured the time interval required for the pulse to reach an object, reflect off of it, and return to be detected. Can air temperature affect the camera’s focus? New cameras use a more reliable infrared system.

Review. A 65.0-kg bungee jumper steps off a bridge with a light bungee cord tied to her body and to the bridge. The unstretched length of the cord is 11.0 m. The jumper reaches the bottom of her motion 36.0 m below the bridge before bouncing back. We wish to find the time interval between her leaving the bridge and her arriving at the bottom of her motion. Her overall motion can be separated into an 11.0-m free fall and a 25.0-m section of simple harmonic oscillation. (a) For the free-fall part, what is the appropriate analysis model to describe her motion? (b) For what time interval is she in free fall? (c) For the simple harmonic oscillation part of the plunge, is the system of the bungee jumper, the spring, and the Earth isolated or nonisolated? (d) From your response in part (c) find the spring constant of the bungee cord. (e) What is the location of the equilibrium point where the spring force balances the gravitational force exerted on the jumper? (f ) What is the angular frequency of the oscillation? (g) What time interval is required for the cord to stretch by 25.0 m? (h) What is the total time interval for the entire 36.0-m drop?

An engineer is designing a triangular weir for measuring the flow rate of a stream of water that has a discharge of 6 cfm. The weir has an included angle of 45$^{\circ}$ and a coefficient of discharge of 0.6. Find the head on the weir.

A simple pendulum with a length of 2.23 m and a mass of 6.74 kg is given an initial speed of 2.06 m/s at its equilibrium position. Assume it undergoes simple harmonic motion. Determine (a) its period, (b) its total energy, and (c) its maximum angular displacement.

For each of the following pdfs let X1, …, Xn be iid observations. Find a complete sufficient statistic, or show that one does not exist. (a) $f(x | \theta)=\frac{2 x}{\theta^{2}}$, 0 < x < $\theta$, $\theta$ > 0 (b) $f(x | \theta)=\frac{\theta}{(1+x)^{1+\theta}}$ , 0 < x < $\infty$, $\theta$ > 0 (c) $f(x | \theta)=\frac{(\log \theta) \theta^{x}}{\theta-1}$, 0 < x < 1, $\theta$ > 1, (d)

$$f(x | \theta)=\left(\begin{array}{l}{2} \\ {x}\end{array}\right) \theta^{x}(1-\theta)^{2-x}$$

, f(x|$\theta$) = e−(x − $\theta$) exp(−e−(x − $\theta$)), $-\infty<x<\infty,-\infty<\theta<\infty$(e) , x = 0, 1, 2, $0 \leq \theta \leq 1$

Water (p=1000 kg/m3) is accelerated from rest in a horizontal pipe that is 80 m long and 30 cm in diameter. If the acceleration rate (toward the downstream end) is 5 m/s2, what is the pressure at the upstream end if the pressure at the downstream end is 90 kPa gage?

In a given city it is assumed that the number of automobile accidents in a given year follows a Poisson distribution. In past years the average number of accidents per year was 15, and this year it was 10. Is it justified to claim that the accident rate has dropped?

Review. Photons of wavelength λ are incident on a metal. The most energetic electrons ejected from the metal are bent into a circular arc of radius R by a magnetic field having a magnitude B. What is the work function of the metal?

A 15.0-mW helium–neon laser emits a beam of circular cross section with a diameter of 2.00 mm. (a) Find the maximum electric field in the beam. (b) What total energyis contained in a 1.00-m length of the beam? (c) Find the momentum carried by a 1.00-m length of the beam.

Solve $\nabla^{2} u=Q(x, y)$ on a rectangle (0 < x < L, 0 < y < H) subject to

$$\ \begin{aligned} &u(0, y)=0, \quad u(x, 0)=0,\\ &u(L, y)=0, \quad u(x, H)=0. \end{aligned}$$

Use a Fourier sine series in y.

According to a study released by the federal Substance Abuse and Mental Health Services Administration (Knight Ridder Tribune, September 9, 1999), approximately $8\%$ of all adult full-time workers are drug users and approximately $70\%$ of adult drug users are employed full-time. a. Is it possible for both of the reported percentages to be correct? Explain. b. Define the events D and E as D = event that a randomly selected adult is a drug user and E = event that a randomly selected adult is employed full-time. What are the estimated values of P(D\ E) and P(E \D) ? c. Is it possible to determine P(D), the probability that a randomly selected adult is a drug user, from the information given? If not, what additional information would be needed?

One mediator of the weak interaction is the Z0 boson, with mass 91 GeV/c2. Use this information to find the order of magnitude of the range of the weak interaction.

In a small city, approximately $15\%$ of those eligible are called for jury duty in any one calendar year. People are selected for jury duty at random from those eligible, and the same individual cannot be called more than once in the same year. What is the probability that an eligible person in this city is selected 2 years in a row? 3 years in a row?