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College Physics by Openstax Chapter 4 Problem 4

Since astronauts in orbit are apparently weightless, a clever method of measuring their masses is needed to monitor their mass gains or losses to adjust diets. One way to do this is to exert a known force on an astronaut and measure the acceleration produced. Suppose a net external force of 50.0 N is exerted and the astronaut’s acceleration is measured to be 0.893 m/s². (a) Calculate her mass. (b) By exerting a force on the astronaut, the vehicle in which they orbit experiences an equal and opposite force. Discuss how this would affect the measurement of the astronaut’s acceleration. Propose a method in which recoil of the vehicle is avoided.

Solution:

We are given the following: $\sum F = 50.0 \ \text{N}$ , and $a=0.893 \ \text{m/s}^{2}$ .

Part A. We can solve for the mass, $m$ by using Newton’s second law of motion.

\begin{align*} \sum F & = ma \\ 50.0 \ \text{N} & = m \left( 0.893 \ \text{m/s}^{2} \right) \\ m & = \frac{50.0 \ \text{N}}{0.893 \ \text{m/s}^{2}} \\ m & = 56.0 \ \text{kg}\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B. The measured acceleration is equal to the sum of the accelerations of the astronauts and the ship. That is

a_{measured}=a_{astronaut}+a_{ship}

If a force acting on the astronaut came from something other than the spaceship, the spaceship would not undergo a recoil. $\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)$

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College Physics by Openstax Chapter 4 Problem 3

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate its acceleration.

Solution:

From Newton’s Second Law of Motion, $\sum F =ma$ . Substituting the given values, we have

\begin{align*} \sum F & = ma \\ 60.0 \ \text{N} & = \left( 4.50 \ \text{kg} \right) \ a \\ a & = \frac{60.0 \ \text{N}}{4.50 \ \text{kg}} \\ a & = 13.3 \ \text{m/s}^{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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College Physics by Openstax Chapter 4 Problem 2

If the sprinter from the previous problem accelerates at that rate for 20 m, and then maintains that velocity for the remainder of the 100-m dash, what will be his time for the race?

Solution:

Solving for the time it takes to reach the first 20 meters.

\begin{align*} \Delta x & =v_{0}t+\frac{1}{2}at^{2} \\ 20 \ \text{m} & = \left( 0 \ \text{m/s} \right)t + \frac{1}{2}\left( 4.20 \ \text{m/s}^{2} \right)t^{2} \\ 20 & = 2.1 t^{2} \\ \frac{20}{2.1}& = \frac{\cancel{2.1} \ t^{2}}{\cancel{2.1}} \\ t^{2} & = 9.5238 \\ \sqrt{t^{2}} & = \sqrt{9.5238} \\ t_{1} & = 3.09 \ \text{s} \end{align*}

We can compute the velocity of the sprinter at the end of the first 20 meters.

\begin{align*} v^2 & = v_{0}^2 + 2ax \\ v^2 & = \left( 0 \ \text{m/s} \right)^2 + 2\left( 4.20 \ \text{m/s}^2 \right) \left( 20 \ \text{m} \right) \\ v & = \sqrt{2\left( 4.20 \right)\left( 20 \right)} \ \text{m/s} \\ v & = 12.96 \ \text{m/s} \end{align*}

For the remaining 80 meters, the sprinter has a constant velocity of 12.96 m/s. The sprinter’s time to run the last 80 meters can be computed as follows.

\begin{align*} \Delta x & = vt \\ 80 \ \text{m} & = \left( 12.96 \ \text{m/s} \right)\ t_{2} \\ \frac{80 \ \text{m}}{12.96 \ \text{m/s}} & =\frac{\cancel{12.96 \ \text{m/s} } \ \ t_{2}}{\cancel{12.96 \ \text{m/s}}} \\ t_{2} & = \frac{80}{12.96} \ \text{s} \\ t_{2} & = 6.17 \ \text{s} \\ \end{align*}

The sprinter’s total time, $t_{T}$ , to finish the 100-m race is the sum of the two times.

\begin{align*} t_{T} & = t_{1} + t_{2} \\ t_{T} & = 3.09 \ \text{s} + 6.17 \ \text{s} \\ t_{T} & = 9.26 \ \text{s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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College Physics by Openstax Chapter 3 Problem 18

You drive 7.50 km in a straight line in a direction 15º east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (This determination is equivalent to find the components of the displacement along the east and north directions.) (b) Show that you still arrive at the same point if the east and north legs are reversed in order.

Solution:

Part A

Consider the illustration shown.

Let D_Ebe the east component of the distance, and D_N be the north component of the distance.

\begin{align*} D_E & = 7.50 \ \sin 15^\circ \\ D_E & = 1.9411 \ \text{km} \\ D_E & = 1.94 \ \text{km} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

\begin{align*} D_N & = 7.50 \ \cos 15^\circ \\ D_N & = 7.2444\ \text{km} \\ D_N & = 7.24 \ \text{km} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Part B

It can be obviously seen from the figure below that you still arrive at the same point if the east and north legs are reversed in order.

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College Physics by Openstax Chapter 3 Problem 17

Repeat Exercise 3.16 using analytical techniques, but reverse the order of the two legs of the walk and show that you get the same final result. (This problem shows that adding them in reverse order gives the same result—that is, B + A = A + B .) Discuss how taking another path to reach the same point might help to overcome an obstacle blocking your other path.

**Figure 3.58** The two displacements A and B add to give a total displacement R having magnitude R and direction θ.

Solution:

Considering the right triangle formed by the vectors A, B, and R. We can solve for the magnitude of R using the Pythagorean Theorem. That is

\begin{align*} R & = \sqrt{A^2+B^2} \\ & = \sqrt{\left( 18.0 \text{m} \right)^2+\left( 25.0 \text{m} \right)^2} \\ & =30.806 \text{m} \\ & \approx 30.8 \text{m} \qquad {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Then we solve for the compass direction by solving the value θ using the same right triangle.

\begin{align*} \theta & = \arctan \left( \frac{B}{A} \right) \\ & = \arctan \left( \frac{25.0 \text{m}}{18.0 \text{m}} \right) \\ & = 54.246 ^\circ \\ & \approx 54.2 ^ \circ \\ \end{align*}

Therefore, the compass direction of the resultant is 54.2° North of West.

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College Physics by Openstax Chapter 3 Problem 16

Solve the following problem using analytical techniques: Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B, as in Figure 3.58, then this problem asks you to find their sum R=A+B.)

Solution:

Considering the right triangle formed by the vectors A, B, and R. We can solve for the magnitude of R using the Pythagorean Theorem. That is

\begin{align*} R & = \sqrt{A^2+B^2} \\ & = \sqrt{\left( 18.0\ \text{m} \right)^2+\left( 25.0\ \text{m} \right)^2} \\ & =30.806 \ \text{m} \\ & \approx 30.8 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Then we solve for the compass direction by solving the value θ using the same right triangle.

\begin{align*} \theta & = \arctan\left( \frac{B}{A} \right) \\ & = \arctan\left( \frac{25.0\ \text{m}}{18.0\ \text{m}} \right) \\ & = 54.246 ^\circ \\ & \approx 54.2 ^ \circ \end{align*}

Therefore, the compass direction of the resultant is 54.2° North of West.

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College Physics by Openstax Chapter 3 Problem 15

Find the north and east components of the displacement from San Francisco to Sacramento shown in Figure 3.57.

Solution:

Consider the following figure.

Using the right triangle formed, we can solve for the east component and the north component. Let S_E be the east component and S_N be the north component of S.

\begin{align*} S_E & = S \cos 45^\circ \\ & = \left( 123 \ \text{km} \right) \cos45^\circ \\ & = 86.974 \ \text{km} \\ & \approx 87.0 \ \text{km} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

\begin{align*} S_N & = S \sin 45^\circ \\ & = \left( 123 \ \text{km} \right) \sin 45^\circ \\ & = 86.974 \ \text{km} \\ & \approx 87.0 \ \text{km} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Therefore, the east and north components are equal at about 87.0 km.

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College Physics by Openstax Chapter 3 Problem 14

Find the following for path D in Figure 3.56: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.

**Figure 3.56** The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.

Solution:

Part A

Looking at path D, we can see that it moves 2 blocks downward, 6 blocks to the right, 4 blocks upward, and 1 block to the left. Thus, the total distance of path D is

\begin{align*} \text{distance} & = \left( 2\times 120\ \text{m} \right)+\left( 6\times 120\ \text{m} \right)+\left( 4\times 120\ \text{m} \right)+\left( 1\times 120\ \text{m} \right) \\ & = 1\ 560 \ \text{m} \\ & = 1.56 \times 10^{3} \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Part B

Looking at the initial and final position of path D, the final position is 5 blocks to the right or 600 meters to the right of the initial position, and 2 blocks or 240 meters upward from the initial position. Refer to the figure below.

Using the right triangle, we can solve for the displacement using the Pythagorean Theorem.

\begin{align*} \text{displacement} & = \sqrt{\left( 600\ \text{m} \right)^2+\left( 240\ \text{m} \right)^2} \\ & = 646.2198 \ \text{m} \\ & \approx 646 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

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College Physics by Openstax Chapter 3 Problem 7

(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction 40.0º north of east (which is equivalent to subtracting B from A —that is, to finding R’=A−B ). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction 40.0º south of west and then 12.0 m in a direction 20.0º east of south (which is equivalent to subtracting A from B —that is, to finding R”=B−A=−R’ ). Show that this is the case.

Solution:

Refer to this problem.

Part A

Consider Figure 3-7A

First, we solve for the value of α using a simple geometry.

\alpha = 40^\circ+\left( 90^\circ-20^\circ \right) = 110^\circ

We can solve for R using the cosine law.

\begin{align*} R^2 & = A^2 +B^2-2AB \cos \alpha \\ R & = \sqrt{A^2 +B^2-2AB \cos \alpha} \\ R & = \sqrt{\left( 12.0\ \text{m} \right)^2+\left( 20.0\ \text{m} \right)^2-2\left( 12.0\ \text{m} \right)\left( 20.0\ \text{m} \right) \cos 110^\circ} \\ R & = 26.6115 \ \text{m} \\ R & = 26.6 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Before we can solve for θ, we need to solve for 𝛽 first using the sine law.

\begin{align*} \frac{\sin \beta}{B} & = \frac{\sin \alpha }{R} \\ \sin \beta & = \frac{B \ \sin \alpha}{R} \\ \beta & = \sin ^{-1} \left( \frac{B \ \sin \alpha}{R} \right) \\ \beta & = \sin ^{-1} \left( \frac{20.0 \ \text{m}\ \sin 110^{\circ}}{26.6115 \ \text{m}} \right) \\ \beta & = 44.9290^\circ \end{align*}

Now, we can solve for θ.

\begin{align*} \theta & = \left( 90^\circ + 20^\circ \right)-\beta \\ \theta & = 110^\circ - 44.9290^\circ \\ \theta & = 65.071 \\ \theta & = 65.1 ^\circ \end{align*}

Therefore, the compass reading is

65.1^\circ, \text{North of East} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}

Part B

Refer to Figure 3-7B

First, we solve for the value of α using a simple geometry.

\alpha = 40^\circ+\left( 90^\circ-20^\circ \right) = 110^\circ

We can solve for R using the cosine law.

\begin{align*} R^2 & = A^2 +B^2-2AB \cos \alpha \\ R & = \sqrt{A^2 +B^2-2AB \cos \alpha} \\ R & = \sqrt{\left( 12.0\ \text{m} \right)^2+\left( 20.0\ \text{m} \right)^2-2\left( 12.0\ \text{m} \right)\left( 20.0\ \text{m} \right) \cos 110^\circ} \\ R & = 26.6115 \ \text{m} \\ R & = 26.6 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Before we can solve for θ, we need to solve for 𝛽 first using the sine law.

\begin{align*} \frac{\sin \beta}{A} & = \frac{\sin \alpha }{R} \\ \sin \beta & = \frac{A \ \sin \alpha}{R} \\ \beta & = \sin ^{-1} \left( \frac{A \ \sin \alpha}{R} \right) \\ \beta & = \sin ^{-1} \left( \frac{12.0 \ \text{m}\ \sin 110^{\circ}}{26.6115 \ \text{m}} \right) \\ \beta & = 25.0708^\circ \end{align*}

Now, we can solve for θ.

\begin{align*} \theta & = 40^\circ + \beta \\ \theta & = 40^\circ + 25.0708^\circ\\ \theta & = 65.0708^\circ\\ \theta & = 65.1 ^\circ \end{align*}

Therefore, the compass reading is

65.1^\circ, \text{South of West} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}

This is consistent with Part A because (A-B) = -(B-A).

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College Physics by Openstax Chapter 2 Problem 64

(a) Take the slope of the curve in Figure 2.64 to find the jogger’s velocity at t=2.5 s. (b) Repeat at 7.5 s. These values must be consistent with the graph in Figure 2.65.

Solution:

Part A

To find the slope at t=2.5 s, we need the position values at t= 0 s and t=5 s. When $t = 0 \ \text{s}$ , $x = 0 \ \text{m}$ , and when $t = 5 \ \text{s}$ , $x = 17.5 \ \text{m}$ . The velocity at t=2.5 s is

\begin{align*} \text{velocity} & =\text{slope} \\ \text{v} & =\frac{\Delta x}{\Delta t} \\ \text{v} & = \frac{x_2-x_1}{t_2-t_1} \\ \text{v} & = \frac{17.5\ \text{m}-0\ \text{m}}{5 \ \text{s}-0\ \text{s}} \\ \text{v} & =3.5 \ \text{m/s} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \\ \end{align*}

Part B

When $t = 10 \ \text{s}$ , $x=2.5 \ \text{m}$ . Considering the points at t=5 s and t=10 s, the slope at 7.5 s is

\begin{align*} \text{velocity} & =\text{slope} \\ \text{v} & =\frac{\Delta x}{\Delta t} \\ \text{v} & = \frac{x_2-x_1}{t_2-t_1} \\ \text{v} & = \frac{2.5\ \text{m}-17.5\ \text{m}}{10 \ \text{s}-5\ \text{s}} \\ \text{v} & =-3.0 \ \text{m/s} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \\ \end{align*}

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Tag Archives: Physics Solutions

College Physics by Openstax Chapter 4 Problem 4

College Physics by Openstax Chapter 4 Problem 3

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate its acceleration.

College Physics by Openstax Chapter 4 Problem 2

If the sprinter from the previous problem accelerates at that rate for 20 m, and then maintains that velocity for the remainder of the 100-m dash, what will be his time for the race?

College Physics by Openstax Chapter 3 Problem 18

College Physics by Openstax Chapter 3 Problem 17

College Physics by Openstax Chapter 3 Problem 16

College Physics by Openstax Chapter 3 Problem 15

Find the north and east components of the displacement from San Francisco to Sacramento shown in Figure 3.57.

College Physics by Openstax Chapter 3 Problem 14

Find the following for path D in Figure 3.56: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.

College Physics by Openstax Chapter 3 Problem 7

College Physics by Openstax Chapter 2 Problem 64

(a) Take the slope of the curve in Figure 2.64 to find the jogger’s velocity at t=2.5 s. (b) Repeat at 7.5 s. These values must be consistent with the graph in Figure 2.65.