Tag Archives: College Physics

Problem 2-8: Motion of land mass around the San Andreas fault

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PROBLEM:

Land west of the San Andreas fault in southern California is moving at an average velocity of about 6 cm/y northwest relative to land east of the fault. Los Angeles is west of the fault and may thus someday be at the same latitude as San Francisco, which is east of the fault. How far in the future will this occur if the displacement to be made is 590 km northwest, assuming the motion remains constant?

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SOLUTION:

From the formula $\overline{v}=\frac{\Delta x}{\Delta t}$ , we can solve for $\Delta t$ as follows

\begin{align*} \Delta t & =\frac{\Delta x}{\overline{v}} \\ \\ & =\frac{5.90 \times 10^{5}\ \text{m}}{6\ \text{cm/year}}\times \frac{100\ \text{cm}}{1\ \text{m}} \\ \\ & = 9.83 \times 10^{6}\ \text{years} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

Therefore, it will take about 9.83 million years.

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Problem 2-7: The rate at which the North American and European continents are moving apart

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PROBLEM:

The North American and European continents are moving apart at a rate of about 3 cm/y. At this rate how long will it take them to drift 500 km farther apart than they are at present?

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SOLUTION:

We know that the formula for velocity is

\text{velocity}=\frac{\text{displacement}}{\text{time}}

If we rearrange the formula and solve for time, we come up with

\text{time}=\frac{\text{displacement}}{\text{velocity}}

Substituting the given values,

\begin{align*} \Delta \text{t} & =\frac{500\:\text{km}}{3\:\text{cm/year}} \\ \\ & =\frac{5\times 10^5\text{ m}}{3\:\text{cm/year}}\times \frac{100\:\text{cm}}{1\:\text{m}} \\ \\ & =1.67\times 10^7\text{years} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\ \end{align*}

Therefore, the time it will take is about $1.67\times 10^{7}\ \text{years}$ .

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Problem 2-6: The average speed and average velocity of a spinning helicopter blade

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PROBLEM:

A helicopter blade spins at exactly 100 revolutions per minute. Its tip is 5.00 m from the center of rotation.

(a) Calculate the average speed of the blade tip in the helicopter’s frame of reference.

(b) What is its average velocity over one revolution?

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SOLUTION:

Part A

The average speed of the blade tip is equal to the distance traveled divided by the time elapsed.

\begin{align*} \text{speed} & =\frac{ \text{distance traveled}}{ \text{time elapsed}} \\ \\ & =\frac{2\pi \text{r}}{ \text{t}} \\ \\ & =\frac{2\pi \left(5.00\text{m}\right)}{60\:\text{s}}\times 100\:\text{rev} \\ \\ & =52.36\:\text{m/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B

After one revolution, the tip of the blade is at the same position as it is originally. This means that the displacement is zero. Thus, the velocity is zero.

\text{v}=0\:\text{m/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

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Problem 2-5: The Earth’s average speed and velocity

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PROBLEM:

(a) Calculate Earth’s average speed relative to the Sun.

(b) What is its average velocity over a period of one year?

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SOLUTION:

Part A

The average speed of the earth is equal to the distance traveled divided by time.

\begin{align*} \text{Average speed of the Earth} & =\frac{\text{Distance Traveled}}{\text{Total time of Travel}} \\ \\ & =\frac{2\pi \text{r}}{\text{t}} \\ \\ & =\frac{2\pi \left(1.50\times 10^{11}\text{ m}\right)}{365.25\:\text{days}}\times \frac{1\:\text{day}}{24\:\text{hours}}\times \frac{1\:\text{hour}}{3600\:\text{sec}} \\ \\ & =2.99\times 10^4\:\text{m/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B

After a period of 1 year, the planet Earth has already returned to its original position with respect to the Sun. This means that we do not have any displacement. Therefore, the average velocity is zero.

\text{Average velocity}=0\:\text{m/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

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Problem 2-4: Distance and Displacement of a given Path D

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PROBLEM:

Find the following for path D in the figure:

(a) The distance traveled.

(b) The magnitude of the displacement from start to finish.

(c) The displacement from start to finish.

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SOLUTION:

Part A

The position of D started from 9 m, then went to 3 m, and then went back to 5 m. The distance traveled by D is the sum of all the paths

\text{distance}=6\:\text{m}+2\:\text{m}=8\:\text{m}\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

Part B

The magnitude of the displacement is the difference between the final position and the initial position WITHOUT regard to sign

\left|\Delta x\right|=\left|9\:\text{m}-5\:\text{m}\right|=4\:\text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

Part C

The displacement is the difference between the final position and the initial position TAKING INTO ACCOUNT the sign

\Delta x=5\:\text{m}-9\:\text{m}=-4\:\text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

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Problem 2-3: Distance and Displacement of a given Path C

PROBLEM:

Find the following for path C in the figure:

(a) The distance traveled.

(b) The magnitude of the displacement from start to finish.

(c) The displacement from start to finish.

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SOLUTION:

Part a

The distance is the sum of all the paths of C.

\text{distance}=8\:\text{m}+2\:\text{m}+2\:\text{m}=12\:\text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

Part b

The magnitude of the displacement is the difference between the final position and the initial position without regard to the sign.

\left|\Delta x\right|=\left|10\:\text{m}-2\:\text{m}\right|=8\:\text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

Part c

The displacement is the difference between the final position and the initial position, taking into account the sign

\Delta x=10\:\text{m}-2\:\text{m}=+8\:\text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

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Problem 2-2: Distance and displacement of a given Path B

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PROBLEM:

Find the following for path B in the figure:

(a) The distance traveled.

(b) The magnitude of the displacement from start to finish.

(c) The displacement from start to finish.

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SOLUTION:

Part a

Based on the figure, B travels from 12 to 7. The distance traveled is 5 meters.

Part b

The magnitude of the displacement is 5 meters.

Part c

The displacement is calculated keeping in mind the sign. The motion started at 12 and ended at 7. Therefore, the displacement is

\Delta x=7\:\text{m}-12\:\text{m}=-5\:\text{meters} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

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Problem 2-1: Distance and displacement for a given path A

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PROBLEM:

Find the following for path A in the figure:

(a) The distance traveled.

(b) The magnitude of the displacement from start to finish.

(c) The displacement from start to finish.

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SOLUTION:

Part A

A travels from 0 to 7. The distance traveled is 7 meters.

Part B

The magnitude of the displacement is 7 meters.

Part C

The displacement is the difference between the final and initial positions.

\begin{align*} \Delta x & =7\:\text{m}-0\:\text{m}=+7\:\text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Problem 1-36: The maximum firing rate of a nerve

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PROBLEM:

Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?

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Solution:

One nerve impulse lasts for 10^-3 s.

\text{max firing rate}=\frac{1\:\text{nerve impulse}}{10^{-3}\:\text{sec}}=10^3\:\text{impulses/sec} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

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Problem 1-35: The approximate number of cells in a hummingbird and in a human

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PROBLEM:

(a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is ten times the mass of a bacterium.

(b) Making the same assumption, how many cells are there in a human?

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SOLUTION:

Part A

The mass of a hummingbird is 10^-2 kg, while the mass of a cell is 10^-15 kg. The number of cells in the hummingbird is

\frac{10^{-2}}{10\left(10^{-15}\right)}=10^{12}\:\text{cells} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

Part B

The mass of a person is 10² kg.

\frac{10^2}{10\left(10^{-15}\right)}=10^{16}\:\text{cells}\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

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