For saving energy, bicycling and walking are far more efficient means of transportation than is travel by automobile. For example, when riding at 10.0 mi/h, a cyclist uses food energy at a rate of about 400 kcal/h above what he would use if merely sitting still. (In exercise physiology, power is often measured in kcal/h rather than in watts. Here 1 kcal = 1 nutritionist’s Calorie = 4 186 J.) Walking at 3.00 mi/h requires about 220 kcal/h. It is interesting to compare these values with the energy consumption required for travel by car. Gasoline yields about $1.30 \times 108$ J/gal. Find the fuel economy in equivalent miles per gallon for a person (a) walking and (b) bicycling.

Under normal conditions the human heart converts about 13.0 J of chemical energy per second into 1.30 W of mechanical power as it pumps blood throughout the body. Metabolizing 1.00 kg of fat can release about $9.00 \times 10^{3}$ Calories of energy. What mass of metabolized fat would power the heart for one day?

A long solenoid, with its axis along the x axis, consists of 200 turns per meter of wire that carries a steady current of 15.0 A. A coil is formed by wrapping 30 turns of thin wire around a circular frame that has a radius of 8.00 cm. The coil is placed inside the solenoid and mounted on an axis that is a diameter of the coil and coincides with the y axis. The coil is then rotated with an angular speed of 4.00$\pi$ rad/s. The plane of the coil is in the yz plane at t = 0. Determine the emf generated in the coil as a function of time.

The cosmic background radiation is blackbody radiation from a source at a temperature of 2.73 K. Use Wien’s law to determine the wavelength at which this radiation has its maximum intensity.

An 80.0-kg skydiver jumps out of a balloon at an altitude of 1 000 m and opens his parachute at an altitude of 200 m. (a) Assuming the total retarding force on the skydiver is constant at 50.0 N with the parachute closed and constant at 3 600 N with the parachute open, find the speed of the skydiver when he lands on the ground. (b) Do you think the skydiver will be injured? Explain. (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00 m/s? (d) How realistic is the assumption that the total retarding force is constant? Explain.

Two transverse sinusoidal waves combining in a medium are described by the wave functions $y1 = 3.00 sin \pi(x + 0.600t) y2 = 3.00 sin \pi(x − 0.600t)$ where x, y1, and y2 are in centimeters and t is in seconds. Determine the maximum transverse position of an element of the medium at (a) x = 0.250 cm, (b) x = 0.500 cm, and (c) x = 1.50 cm. (d) Find the three smallest values of x corresponding to antinodes.

The two speakers of a boom box are 35.0 cm apart. A single oscillator makes the speakers vibrate in phase at a frequency of 2.00 kHz. At what angles, measured from the perpendicular bisector of the line joining the speakers, would a distant observer hear maximum sound intensity? Minimum sound intensity? (Take the speed of sound as 340 m/s.)

A boy in a wheelchair (total mass 47.0 kg) has speed 1.40 m/s at the crest of a slope 2.60 m high and 12.4 m long. At the bottom of the slope his speed is 6.20 m/s. Assume air resistance and rolling resistance can be modeled as a constant friction force of 41.0 N. Find the work he did in pushing forward on his wheels during the downhill ride.

Show that $|R|^{2}+|T|^{2}=1$ for the delta function potential in the text.

Given that two objects having masses equal to $3.00 \mathrm{~kg}$ and $5.00 \mathrm{~kg}$ are connected by a light string that passes over a light frictionless pulley to form an Atwood machine as shown in Active Figure given in the textbook. Calculate the tension in the string.

Expand $(\vec{V} \cdot \nabla) \vec{V}$ in rectangular coordinates by direct substitution of the velocity vector to obtain the convective acceleration of a fluid particle. Verify the results given in Eqs. 5.11. $a_{x_{p}}=\frac{D u}{D t}=u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+w \frac{\partial u}{\partial z}+\frac{\partial u}{\partial t}$, $a_{y_{p}}=\frac{D v}{D t}=u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}+w \frac{\partial v}{\partial z}+\frac{\partial v}{\partial t}$, $a_{z_{p}}=\frac{D w}{D t}=u \frac{\partial w}{\partial x}+v \frac{\partial w}{\partial y}+w \frac{\partial w}{\partial z}+\frac{\partial w}{\partial t}$.

A rectangular container, of base dimensions $0.4 \mathrm{m} \times 0.2 \mathrm{m}$ and height 0.4 m, is filled with water to a depth of 0.2 m; the mass of the empty container is 10 kg. The container is placed on a plane inclined at $30^{\circ}$ to the horizontal. If the coefficient of sliding friction between the container and the plane is 0.3, determine the angle of the water surface relative to the horizontal.

For the functions, do the following: Find any inflection points off f. f(x)=x+sin x $(0 \leq x \leq 2 \pi)$

For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate. Do not evaluate the integrals. $\int x^{2} \ln \left(x^{3}+1\right) d x$