FIGURE shows four electric charges located at the comers

a. How many days will it take a spaceship to accelerate to the speed of light \(\left(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\) with the acceleration g? b. How far will it travel during this interval? c. What fraction of a light year is your answer to part b? A light year is the distance light travels in one year. NOTE We know, from Einstein's theory of relativity, that no object can travel at the speed of light. So this problem, while interesting and instructive, is not realistic. Equation Transcription: Text Transcription: (3.0 x 10^8)

Problem 41P A flock of ducks is trying to migrate south for the winter, but they keep being blown off course by a wind blowing from the west at 6.0 m/s. A wise elder duck finally realizes that the solution is to fly at an angle to the wind. If the ducks can fly at 8.0 m/s relative to the air, what direction should they head in order to move directly south?

If \(\vec{C}=\vec{A}+\vec{B}\), can \(C=A+B ? \operatorname{Can} C>A+B\)? For each, show how or explain why not. Equation Transcription: Text Transcription: ^vecC=^vecA+^vecB C=A+B C>A+B

Section 3.1 Vectors Section 3.2 Properties of Vectors Trace the vectors in Figure EX3.1 onto your paper. Then find \(\text { (a) } \vec{A}+\vec{B} \text { and (b) } \vec{A}-\vec{B} \text {. }\) Equation Transcription: Text Transcription: ^vecA+^vecB ^vecA-^vecB

a. Suppose you are driving at speed \(v_{0}\) when a sudden obstacle in the road forces you to make a quick stop. If your reaction time before applying the brakes is \(t_{R}\), what constant deceleration (absolute value of \(a_{\mathrm{x}}\) ) do you need to stop in distance \(d\)? Assume that \(d\) is larger than the car travels during your reaction time. b. Suppose you are driving at \(21 \mathrm{~m} / \mathrm{s}\) when you suddenly see an obstacle \(50 \mathrm{~m}\) ahead. If your reaction time is \(0.50 \mathrm{~s}\) and if your car's maximum deceleration is \(6.0 \mathrm{~m} / \mathrm{s}^{2}\), can you stop in time to avoid a collision? Equation Transcription: Text Transcription: v_0 t_R a_x d d 21 m/s 50 m 0.50 s 6.0 m/s^2

Section 3.1 Vectors Section 3.2 Properties of Vectors Trace the vectors in Figure EX3.2 onto your paper. Then find (a) \(\vec{A}+\vec{B}\) and (b) \(\vec{A}-\vec{B}\) Equation Transcription: Text Transcription: ^vec A+^vec B ^vec A+^vec B

Three forces are exerted on an object placed on a tilted floor in Figure P3.43. The forces are measured in newtons (N). Assuming that forces are vectors, a. What is the component of the net force \(\vec{F}_{\text {net }}=\vec{F}_{1}+\vec{F}_{2}+\vec{F}_{3}\) parallel to the floor? b. What is the component of \(\vec{F}_{\text {net }}\) perpendicular to the floor? c. What are the magnitude and direction of \(\vec{F}_{\text {net }}\)? Equation Transcription: Text Transcription: ^vec F_net=^vec F_1+^vec F_2+^vec F_3 ^vec F_net ^vec F_net

\(\text { If } \vec{C}=\vec{A}+\vec{B}, \text { can } C=0 ? \text { Can } C<0 ? \text { For each, show how or explain why not. }\) Equation Transcription: Text Transcription: ^vec C=^vec A+^vec B C=0 C<0

Problem 4CQ Is it possible to add a scalar to a vector? If so, demonstrate. If not, explain why not.

Problem 4E Section 3.3 Coordinate Systems and Vector Components A velocity vector 40° below the positive x -axis has a y -component of ?10 m/s. What is the value of its x -component?

Problem 6CQ Can a vector have a component equal to zero and still have nonzero magnitude? Explain.

\(\text { How would you define the zero vector } \overrightarrow{0} \text { ? }\) Equation Transcription: Text Transcription: ^vec 0

Problem 5E Section 3.3 Coordinate Systems and Vector Components A position vector in the first quadrant has an x -component of 8 m and a magnitude of 10 m. What is the value of its y -component?

Problem 7CQ Can a vector have zero magnitude if one of its components is nonzero? Explain.

a. What are the \(x\) - and \(y\)-components of vector \(\vec{E}\) shown in FIGURE EX3.3 in terms of the angle \(\theta\) and the magnitude \(E\) ? b. For the same vector, what are the \(x\) - and \(y\)-components in terms of the angle \(\phi\) and the magnitude \(E\) ? Equation Transcription: Text Transcription: x- y-components ^vec E phi E x- y-components phi E

Draw each of the following vectors, then find its \(x\)- and \(y\)-components. a. \(\vec{r}=(100 \mathrm{~m}, 45\) below positive \(x\)-axis ) b. \(\vec{v}=(300 \mathrm{~m} / \mathrm{s}, 20\) above positive \(x\)-axis) c. \(\vec{a}=(5.0 \mathrm{~m} / \mathrm{s} 2\) negative \(y\)-direction) Equation Transcription: Text Transcription: x- and y-components ^vec r=(100 m, 45 below positive x-axi) ^vec v=(300 m/s, 20 above positive x-axis) ^vec a=(5.0 m/s^2,negative y-direction)

Draw each of the following vectors, then find its \(x\) - and \(y\)-components. a. \(\vec{r}=(10 \mathrm{~m} / \mathrm{s}\), negative \(y\)-direction) b. \(\vec{a}=\left(20 \mathrm{~m} / \mathrm{s}^{2}, 30^{\circ}\right\) below positive \(x\)-axis) c. \(\vec{F}=\left(100 N, 36.9^{\circ}\right.\) counterclockwise from positive \(y\)-axis) Equation Transcription: Text Transcription: x- and y-components ^vec r=(10m/s, negative y-direction) ^vec a=(20 m/s^2, 30^circ below positive x-axis ^vec F=(100 N, 36.9^circ counterclockwise from postive y-axis

Let \(\vec{C}=\left(3.15 \mathrm{~m}, 15^{\circ}\right.\) above the negative \(x\)-axis) and \(\vec{D}=\left(25.6 \mathrm{~m}, 30^{\circ}\right.\) to the right of the negative \(y\)-axis). Find the magnitude, the \(x\)-component, and the \(y\)-component of each vector. Equation Transcription: Text Transcription: ^vec C=(3.15 m, 15^circ x-axis ^vec D=(25.6 m, 30^circ y-axis x-components y-components

Problem 8CQ Suppose two vectors have unequal magnitudes. Can their sum be zero? Explain.

Problem 9CQ Are the following statements true or false? Explain your answer. a. The magnitude of a vector can be different in different coordinate systems. ________________ b. The direction of a vector can be different in different coordinate systems. ________________ c. The components of a vector can be different in different coordinate systems.

Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. a. \(\vec{B}=-4 \hat{\imath}+4 \hat{\jmath}\) b. \(\vec{r}=(-2.0 \hat{\imath}-1.0 \hat{\jmath}) \mathrm{cm}\) c. \(\vec{v}=(-10 \hat{\imath}-100 \hat{\jmath}) \mathrm{m} / \mathrm{s}\) d. \(\vec{a}=(20 \hat{\imath}+10 \hat{\jmath}) \mathrm{m} / \mathrm{s}^{2}\) Equation Transcription: Text Transcription: ^vec B=-4^hat i+4^hat ^vec r=(-2.0^hat i-1.0^hat j)cm ^vec v=(-10^hat i-100^hat j)m/s ^vec a=(20^hat i+10^hat j)m/s^2

The magnetic field inside an instrument is \(\vec{B}=(2.0 \hat{\imath}-1.0 \hat{\jmath}) \mathrm{T}\) where \(\vec{B}\) represents the magnetic field vector and \(\mathrm{T}\) stands for tesla, the unit of the magnetic field. What are the magnitude and direction of the magnetic field? Equation Transcription: Text Transcription: ^vec B=(2.0^hat i-1.0^hat j) ^vec B T

Let \(\vec{A}=2 \hat{\imath}+3 \hat{\jmath}\) and \(\vec{B}=4 \hat{\imath}-2 \hat{\jmath}\). a. Draw a coordinate system and on it show vectors \(\vec{A}\) and \(\vec{B}\). b. Use graphical vector subtraction to find \(\vec{C}=\vec{A}-\vec{B}\). Equation Transcription: Text Transcription: ^vec A=2 ^hat i+3^vec j ^vec B=4^vec i-2^vec j ^vec A ^vec B ^vec C=^vec A-B

Draw each of the following vectors, label an angle that specifies the vector's direction, then find the vector's magnitude and direction. a. \(\vec{A}=4 \hat{\imath}-6 \hat{\jmath}\) b. \(\vec{r}=(50 \hat{\imath}+80 \hat{\jmath}) \mathrm{m}\) c. \(\vec{v}=(-20 \hat{\imath}+40 \hat{\jmath}) \mathrm{m} / \mathrm{s}\) d. \(\vec{a}=(2.0 \hat{\imath}-6.0 \hat{\jmath}) \mathrm{m} / \mathrm{s}^{2}\) Equation Transcription: Text Transcription: ^vec A=4^hat i-6^hat j ^vec r=(50^hat i+80^hat j)m ^vec v=(-20^hat i+40^hat j)m/s ^vec A=(2.0^hat i-6.0^hat j)m/s^2

Let \(\vec{A}=4 \hat{\imath}-2 \hat{\jmath}, \vec{B}=-3 \hat{\imath}+5 \hat{\jmath}\), and \(\vec{C}=\vec{A}+\vec{B}\). a. Write vector \(\vec{C}\) in component form. b. Draw a coordinate system and on it show vectors \(\vec{A}, \vec{B}\), and \(\vec{C}\). c. What are the magnitude and direction of vector \(\vec{C}\) ? Equation Transcription: Text Transcription: ^vec A=4 ^hat i-2 ^hat j, B=3 ^hat i+5 ^hat j ^vec C=^vec A+^vec B Cu ^vec A, ^vec B, ^vec C ^vec C

Let \(\vec{A}=4 \hat{\imath}-2 \hat{\jmath}, \vec{B}=-3 \hat{\imath}+5 \hat{\jmath}\), and \(\vec{E}=4 \vec{A}+2 \vec{B}\). a. Write vector \(\vec{E}\) in component form. b. Draw a coordinate system and on it show vectors \(\vec{A}, \vec{B}\), and \(\vec{E}\). c. What are the magnitude and direction of vector \(\vec{E}\)? Equation Transcription: Text Transcription: ^vec A=4 ^hat i-2 ^hat j, B=3 ^hat i+5 ^hat j ^vec E=4^vec A+2^vec B ^vec E ^vec A,^vec B, ^vec E ^vec E

Let \(\vec{A}=4 \hat{\imath}-2 \hat{\jmath}, \vec{B}=-3 \hat{\imath}+5 \hat{\jmath}\), and \(\vec{D}=\vec{A}-\vec{B}\). a. Write vector \(\vec{D}\) in component form. b. Draw a coordinate system and on it show vectors \(\vec{A}, \vec{B}\), and \(\vec{D}\). c. What are the magnitude and direction of vector \(\vec{D}\)? Equation Transcription: Text Transcription: ^vec A=4 ^hat i-2 ^hat j,B=3 ^hat i+5 ^hat j ^vec D=^vec A-^vec B ^vec D ^vec A,^vec B,^vec D

Let \(\vec{B}=\left(5.0 \mathrm{~m}, 60^{\circ}\right.\) counterclockwise from vertical). Find the \(x\) - and \(y\)-components of \(\vec{B}\) in each of the two coordinate systems shown in FIGURE EX3.17. Equation Transcription: Text Transcription: ^vec B=(5.0 m, 60^circ x- y-components ^vec B

Let \(\vec{E}=2 \hat{\imath}+3 \hat{\jmath}\) and \(\vec{F}=2 \hat{\imath}-2 \hat{\jmath}\). Find the magnitude of a. \(\vec{E}\) and \(\vec{F}\) b. \(\vec{E}+\vec{F}\) c. \(-\vec{E}-2 \vec{F}\) Equation Transcription: Text Transcription: ^vec E=2 ^hat i+3j Fu=2 ^hat i-2 ^hat j ^vec E ^vec F ^vecE+^vecF -^vec E-2^vec F

Let \(\vec{A}=4 \hat{\imath}-2 \hat{\jmath}, \vec{B}=-3 \hat{\imath}+5 \hat{\jmath}\), and \(\vec{F}=\vec{A}-4 \vec{B}\). a. Write vector \(\vec{F}\) in component form. b. Draw a coordinate system and on it show vectors \(\vec{A}, \vec{B}\), and \(\vec{F}\). c. What are the magnitude and direction of vector \(\vec{F}\)? Equation Transcription: Text Transcription: ^vec A=4 ^hat i-2 ^hat j,B=-3 ^hat i+5 ^hat j ^vec F=^vec A-4^vec B ^vec F ^vec A,^vec B,^vec F ^vec F

What are the \(x\)- and \(y\)-components of the velocity vector shown in Figure EX3.18? Equation Transcription: Text Transcription: x- y-components

The position of a particle as a function of time is given by \(\vec{r}=(5.0 \hat{\imath}+4.0 \hat{\jmath}) t^{2} \mathrm{~m}\), where \(t\) is in seconds. a. What is the particle's distance from the origin at \(t=0,2\), and \(5 \mathrm{~s}\)? b. Find an expression for the particle's velocity \(\vec{v}\) as a function of time. c. What is the particle's speed at \(t=0,2\), and \(5 \mathrm{~s}\)? Equation Transcription: Text Transcription: ^vec r=(5.0 ^hat i+4.0 ^hat j)t^2 m t t=0, 2, 5 s ^vec v t=0, 2, 5 s

Let \(\vec{A}=\left(3.0 \mathrm{~m}, 20^{\circ}\right.\) south of east), \(\vec{B}=(2.0 \mathrm{~m}\), north), and \(\vec{C}=\left(5.0 \mathrm{~m}, 70^{\circ}\right.\) south of west). a. Draw and label \(\vec{A}, \vec{B}\), and \(\vec{C}\) with their tails at the origin. Use a coordinate system with the \(x$-axis to the east. b. Write \(\vec{A}, \vec{B}\), and \(\vec{C}\) in component form, using unit vectors. c. Find the magnitude and the direction of \(\vec{D}=\vec{A}+\vec{B}+\vec{C}\). Equation Transcription: Text Transcription: ^vec A=(3.0 m, 20^circ south of east),B=(2.0 m, north) ^vec C(5.0 m, 70^circ south of west) ^vec A,^vec B,^vec C x-axis ^vec A,^vec B,^vec C ^vec D=^vec A+^vec B+^vec C

For the three vectors shown in FIGURE P3.23, \(\vec{A}+\vec{B}+\vec{C}=1 \hat{\jmath}\). What is vector \(\vec{B}\)? a. Write \(\vec{B}\) in component form. b. Write \(\vec{B}\) as a magnitude and a direction. Equation Transcription: Text Transcription: ^vec A+^vec B+^vec C=1 ^hatj ^vec B ^vec B ^vec B

FIGURE P3.26 shows vectors \(\vec{A}\) and \(\vec{B}\). Find \(\vec{D}=2 \vec{A}+\vec{B}\). Write your answer in component form. Equation Transcription: Text Transcription: ^vec A ^vec B D u=2^vec A+^vec B

Figure P3.25 shows vectors \(\vec{A}\) and \(\vec{B}\). Find vector \(\vec{C}\) such that \(\vec{A}+\vec{B}+\vec{C}=\overrightarrow{0}\). Write your answer in component form. Equation Transcription: Text Transcription: ^vec A ^vec B ^vec C ^vec A+^vec B+^vec C=^vec 0

Carlos runs with velocity \(\vec{v}=\left(5.0 \mathrm{~m} / \mathrm{s}, 25^{\circ}\right.\) north of east) for 10 minutes. How far to the north of his starting position does Carlos end up? Equation Transcription: Text Transcription: ^vec v=(5.0 m/s, 25 ^circ north of east)

FIGURE P3.22 shows vectors \(\vec{A}\) and \(\vec{B}\). Let \(\vec{C}=\vec{A}+\vec{B}\). a. Reproduce the figure on your page as accurately as possible, using a ruler and protractor. Draw vector \(\vec{C}\) on your figure, using the graphical addition of \(\vec{A}\) and \(\vec{B}\). Then determine the magnitude and direction of \(\vec{C}\) by measuring it with a ruler and protractor. b. Based on your figure of part a, use geometry and trigonometry to calculate the magnitude and direction of \(\vec{C}\). c. Decompose vectors \(\vec{A}\) and \(\vec{B}\) into components, then use these to calculate algebraically the magnitude and direction of \(\vec{C}\). Equation Transcription: Text Transcription: ^vec A ^vec B ^vec C=^vec A+^vec B ^vec C ^vec A ^vec B ^vec C ^vec C ^vec A ^vec B Cu

Problem 27P Find a vector that points in the same direction as the vector (î + ? ) and whose magnitude is 1.

For the three vectors shown in FIGURE P3.23, \(\vec{A}+\vec{B}+\vec{C}=1 \hat{\jmath}\). What is vector \(\vec{B}\)? a. Write \(\vec{B}\) in component form. b. Write \(\vec{B}\) as a magnitude and a direction. Equation Transcription: Text Transcription: ^vec A+^vec B+^vec C=1 ^hat j ^vec B ^vec B ^vec B

If While vacationing in the mountains you do some hiking. In the morning, your displacement is \(\vec{S}_{\text {morning }}=(2000 \mathrm{~m}\), east \()+\) \((3000 \mathrm{~m}\), north \()+(200 \mathrm{~m}\), vertical). After lunch, your displacement is \(\vec{S}_{\text {afternoon }}=(1500 \mathrm{~m}\), west \()+(2000 \mathrm{~m}\), north \()-\) \((300 \mathrm{~m}\), vertical). a. At the end of the hike, how much higher or lower are you compared to your starting point? b. What is the magnitude of your net displacement for the day? Equation Transcription: Text Transcription: ^vec S_morning=(2000 m, east)+(3000 m, north)+(200 m, vertical) ^vec S_afternoon=(1500 m, west)+(2000 m, north)-(300 m, vertical)

Problem 31P Bob walks 200 m south, then jogs 400 m southwest, then walks 200 m in a direction 30° east of north. a. Draw an accurate graphical representation of Bob’s motion. Use a ruler and a protractor! ________________ b. Use either trigonometry or components to find the displacement that will return Bob to his starting point by the most direct route. Give your answer as a distance and a direction. ________________ c. Does your answer to part b agree with what you can measure on your diagram of part a?

Problem 35P Jack and Jill ran up the hill at 3.0 m/s. The horizontal component of Jill’s velocity vector was 2.5 m/s. a. What was the angle of the hill? b. What was the vertical component of Jill’s velocity?

Problem 30P The minute hand on a watch is 2.0 cm in length. What is the displacement vector of the tip of the minute hand a. From 8:00 to 8:20 A.M.? ________________ b. From 8:00 to 9:00 A.M.?

Problem 32P Jim’s dog Sparky runs 50 m northeast to a tree, then 70 m west to a second tree, and finally 20 m south to a third tree. a. Draw a picture and establish a coordinate system. ________________ b. Calculate Sparky’s net displacement in component form. ________________ c. Calculate Sparky’s net displacement as a magnitude and an angle.

Problem 33P A field mouse trying to escape a hawk runs east for 5.0 m, darts southeast for 3.0 m, then drops 1.0 m straight down a hole into its burrow. What is the magnitude of its net displacement?

Problem 34P A cannon tilted upward at 30° fires a cannonball with a speed of 100 m/s. At that instant, what is the component of the cannonball’s velocity parallel to the ground?

Problem 37P Mary needs to row her boat across a 100-m-wide river that is flowing to the east at a speed of 1.0 m/s. Mary can row the boat with a speed of 2.0 m/s relative to the water. a. If Mary rows straight north, how far downstream will she land? ________________ b. Draw a picture showing Mary’s displacement due to rowing, her displacement due to the river’s motion, and her net displacement.

Problem 36P A pine cone falls straight down from a pine tree growing on a 20° slope. The pine cone hits the ground with a speed of 10 m/s. What is the component of the pine cone’s impact velocity (a) parallel to the ground and (b) perpendicular to the ground?

The treasure map in Figure P3.38 gives the following directions to the buried treasure: “Start at the old oak tree, walk due north for \(500\ paces\), then due east for \(100\ paces\). Dig.” But when you arrive, you find an angry dragon just north of the tree. To avoid the dragon, you set off along the yellow brick road at an angle 60 east of north. After walking \(300\ paces\) you see an opening through the woods. Which direction should you go, and how far, to reach the treasure? Equation Transcription: Text Transcription: 500 paces 100 paces 300 paces

Problem 39P A jet plane is flying horizontally with a speed of 500 m/s over a hill that slopes upward with a 3% grade (i.e., the “rise” is 3% of the “run”). What is the component of the plane’s velocity perpendicular to the ground?

The bacterium E. coli is a single-cell organism that lives in the gut of healthy animals, including humans. When grown in a uniform medium in the laboratory, these bacteria swim along zigzag paths at a constant speed of \(20 mm/s\). Figure P3.40 shows the trajectory of an E. coli as it moves from point A to point E. What are the magnitude and direction of the bacterium’s average velocity for the entire trip? Equation Transcription: Text Transcription: 20 mm/s