Ch1-HW1-solutions - schopper (as55864) – Ch1-HW1 –...

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Unformatted text preview: schopper (as55864) – Ch1-HW1 – Antoniewicz – (56465) This print-out should have 36 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 (part 1 of 5) 2.0 points Consider vectors A and B such that A = 100, −600, −700 and 1 003 (part 3 of 5) 2.0 points Find A . Correct answer: 927.362. Explanation: We follow the same procedure as for the previous part of the problem. B = −400, 400, 600 . Find A + B . 1. 300, −200, −300 2. 100, −300, 100 3. 300, 200, −100 4. −300, −200, −100 correct 5. 0, 100, 300 Explanation: To add vectors, we add respective components: Ax + Bx = 100 + (−400) = −300 , Ay + By = −600 + 400 = −200 , and Az + Bz = −700 + 600 = −100 , so A + B = Ax + B x , Ay + B y , Az + B z = −300, −200, −100 . 002 (part 2 of 5) 2.0 points Find A + B . Correct answer: 374.166. Explanation: The magnitude of A + B is A+B = (Ax + Bx )2 +(Ay + By )2 +(Az + Bz )2 = (−300)2 + (−200)2 + (−100)2 = 374.166 . A= = A2 + A2 + A2 x y z (100)2 + (−600)2 + (−700)2 = 927.362 . 004 (part 4 of 5) 2.0 points Find B . Correct answer: 824.621. Explanation: We follow the same procedure again. B= = 2 2 2 Bx + By + Bz (−400)2 + (400)2 + (600)2 = 824.621 . 005 (part 5 of 5) 2.0 points Find A + B . Correct answer: 1751.98. Explanation: Here we simply add the values we obtained in parts 3 and 4: A + B = 927.362 + 824.621 = 1751.98 . 006 10.0 points Consider the following figure: schopper (as55864) – Ch1-HW1 – Antoniewicz – (56465) r s t Which of the following statements about the three vectors in the figure are correct? List all that apply, separated by commas. and option A is true. Following the same procedure for the remaining options, we can determine that A, B, and E are true statements, while the others are false. 007 (part 1 of 3) 3.333 points A unit vector v lies in the xy plane, at an angle of 105◦ from the +x axis, with a positive y component. What are the components of the unit vector v = vx , ˆy , ˆz ? ˆ ˆvv A s=t−r 105◦ B r =t−s C r+t=s v D s+t=r E r+s=t Find vx . ˆ Correct answer: A, B, E. Explanation: Vector subtraction can be tricky. Just as one way to subtract scalar quantities is to add the negative of the number being subtracted (i.e., instead of 5 − 3 = 2 we could write 5 + (−3) = 2), we can do the same with vectors. We start by writing down the first vector. For option A, the first vector in the subtraction is t, so we draw it: Then, we add the negative of the vector being subtracted. When we write down −r , it will be pointing in the opposite direction from r. We place −r ’s starting point at the tip of t, and from here, we are simply adding vectors: t s Correct answer: −0.258819. Explanation: cos 105◦ sin 105◦ 105◦ v The x component of v is given by ˆ t vx = cos 105◦ = −0.258819. ˆ 008 (part 2 of 3) 3.333 points Find vy . ˆ Correct answer: 0.965926. −r Explanation: The y component of v is given by ˆ vy = sin 105◦ = 0.965926. ˆ Notice that the resultant vector, s, is exactly the same as the original s in the figure we started with. So t + (−r ) = t − r = s, 2 009 (part 3 of 3) 3.333 points Find vz . ˆ Correct answer: 0. schopper (as55864) – Ch1-HW1 – Antoniewicz – (56465) Explanation: This vector lies in the xy plane, so it has no z component. Therefore vz = 0. ˆ 010 (part 1 of 6) 1.667 points In the following figure, three vectors are represented by arrows in the xy plane. Each square in the grid represents one meter. Determine the components of each vector (an accuracy of 0.5 m is fine), and then calculate the magnitude of the vector. What is A ? Give your answer in units of meters. Correct answer: 6.18466 m. Explanation: We use the Pythagorean theorem: A= = A2 + A2 + A2 x y z (−6 m)2 + (+1.5 m)2 + (0 m)2 = A 3 √ 38.25 m2 = 6.18466 m. B C 012 (part 3 of 6) 1.667 points Which choice best represents the components of B = Bx , By , Bz ? 1. −3 m, +4.5 m, 0 m Which choice best represents the components of A = Ax , Ay , Az ? 1. −6 m, +1.5 m, 0 m correct 2. −5.5 m, +1 m, 0 m 3. +6 m, −1.5 m, 0 m 4. −6 m, −1.5 m, 0 m 5. +5.5 m, +1.5 m, 0 m Explanation: To find the components of A, we just count the units by which A stretches in the x and y directions, letting the direction determine the sign. A stretches 6 units along the x-direction, and the direction is to the left. Since the units are meters, we know that Ax = −6 m. Similarly for the y direction, A stretches 1.5 units upward, meaning Ay = +1.5 m. Since the vector lies in the xy plane, the third component Az = 0. 011 (part 2 of 6) 1.667 points 2. −3 m, −4 m, 0 m 3. +3 m, −4.5 m, 0 m correct 4. +3.5 m, −4.5 m, 0 m 5. +3.5 m, −4 m, 0 m Explanation: B stretches 3 units along the positive xdirection, so Bx = +3 m. B stretches 4.5 units downward, meaning By = −4.5 m. Since the vector lies in the xy plane, the third component Bz = 0. 013 (part 4 of 6) 1.667 points What is B ? Give your answer in units of meters. Correct answer: 5.40833 m. Explanation: The procedure is the same as for part 2: B= 2 2 2 Bx + By + Bz schopper (as55864) – Ch1-HW1 – Antoniewicz – (56465) = (+3 m)2 + (−4.5 m)2 + (0 m)2 √ = 29.25 m2 = 5.40833 m. 014 (part 5 of 6) 1.667 points Which choice best represents the components of C = Cx , Cy , Cz ? 1. +2.5 m, −1.5 m, 0 m 2. +2.5 m, +1.5 m, 0 m correct 4 A star is located at S = 2 × 1010 , −3 × 1010 , 5 × 1010 . What is R, the vector pointing from the star to the planet? 1. R = 5 × 1010 , −1.2 × 1011 , 6 × 1010 2. R = 8 × 1010 , −9 × 1010 , 6 × 1010 3. R = 8 × 1010 , −9 × 1010 , 7 × 1010 4. R = 4 × 1010 , −7 × 1010 , 1 × 1011 correct 3. +2.5 m, +1 m, 0 m 4. +2 m, +1.5 m, 0 m 5. −2.5 m, +1.5 m, 0 m Explanation: C stretches 2.5 units along the positive xdirection, so Cx = +2.5 m. C stretches 1.5 units upward, meaning Cy = +1.5 m. Since the vector lies in the xy plane, the third component Cz = 0. 015 (part 6 of 6) 1.667 points What is C ? Give your answer in units of meters. Correct answer: 2.91548 m. Explanation: We use the Pythagorean theorem: C= = 2 2 2 Cx + Cy + Cz (+2.5 m)2 + (+1.5 m)2 + (0 m)2 √ = 8 . 5 m2 5. R = 4 × 1010 , −1.1 × 1011 , 9 × 1010 Explanation: This is a vector subtraction problem. To find R, we subtract S − P by respective components: Sx − Px = 2 × 1010 − (−2 × 1010 ) = 4 × 1010 Sy − Py = −3 × 1010 + 4 × 1010 = −7 × 1010 Sz − Pz = 5 × 1010 + (−5 × 1010 ) = 1 × 1011 So R=S−P = 4 × 1010 , −7 × 1010 , 1 × 1011 . 017 (part 2 of 5) 2.0 points What is R ? Correct answer: 1.28452 × 1011 . Explanation: To find R , we use the Pythagorean theorem. = 2.91548 m. R= 016 (part 1 of 5) 2.0 points A planet is located at P = −2 × 1010 , 4 × 1010 , −5 × 1010 . (4 × 1010 )2 + (−7 × 1010 )2 + (1 × 1011 )2 = 1.65 × 1022 = 1.28452 × 1011 . schopper (as55864) – Ch1-HW1 – Antoniewicz – (56465) 018 (part 3 of 5) 2.0 points For the remaining three parts of this problem, ˆ you will find the components of R, the unit vector in the direction of R. Begin by finding ˆ Rx . Correct answer: 0.3114. Explanation: We simply divide Rx by the magnitude: 4 × 1010 ˆ x = rx = R = 0.3114. 1.28452 × 1011 R ˆ by finding A and the components of A. Find A . Correct answer: 469.042 m/s2 . Explanation: To find the magnitude of the vector A, we use the Pythagorean theorem: A= = Correct answer: −0.544949. Explanation: We divide Ry by the magnitude: Ry −7 × 1010 ˆ = −0.544949. = Ry = 1.28452 × 1011 R 020 (part 5 of 5) 2.0 points ˆ z. Find R Correct answer: 0.778499. Explanation: We divide Rz by the magnitude: 1 × 1011 Rz ˆ = = 0.778499. Rz = 1.28452 × 1011 R 021 (part 1 of 4) 2.5 points Write the vector A = 200, 300, −300 m/s2 as the product ˆˆˆ ˆ A · A = A · Ax , Ay , Az A2 + A2 + A2 x y z (200)2 + (300)2 + (−300)2 = 019 (part 4 of 5) 2.0 points ˆ y. Find R 5 2.2 × 105 = 469.042 m/s2 . 022 (part 2 of 4) 2.5 points ˆx . Find A Correct answer: 0.426401 m/s2 . Explanation: To find the unit vector, we use the formula Ax , Ay , Az A ˆ A= = , A A so we can use our answer from part 1 and simply divide each component by the magnitude. ˆ For Ax , therefore, we get Ax 200 ˆ Ax = = = 0.426401 m/s2 . 469.042 m/s2 A 023 (part 3 of 4) 2.5 points ˆy . Find A Correct answer: 0.639602 m/s2 . Explanation: Following the procedure from part 2, we get Ay 300 ˆ = 0.639602 m/s2 . = Ay = 2 469.042 m/s A schopper (as55864) – Ch1-HW1 – Antoniewicz – (56465) 024 (part 4 of 4) 2.5 points ˆz Find A Correct answer: −0.639602 m/s2 . Correct answer: −104. Explanation: One more time: Explanation: Following the procedure from part 2 again, we get Az −300 ˆ Az = = −0.639602 m/s2 . = 469.042 m/s2 A 025 (part 1 of 3) 3.333 points The vector a = 0.03, −1.8, 26.0 and the scalar f = −4. Let b = f a. What are the components of b = bx , by , bz ? First, find bx . Correct answer: −0.12. Explanation: Multiplying a vector by a scalar means multiplying each component by that scalar. So bx = f × a x = −4 × 0.03 = −0.12 . bz = f × a z = −4 × 26.0 = −104 . 028 (part 1 of 4) 2.5 points A man is standing on the roof of a building with his head at the position r m = 12 m, 32 m, 13 m . He sees the top of a tree, which is at the position r t = −24 m, 33 m, 43 m . What are the components of the relative position vector r tm that points from the man’s head to the top of the tree? Start by finding tm rx . Answer in m. Correct answer: −36 m. Explanation: We simply subtract the two x components, starting with the position where we want the vector to point, which in this case is the tree: m tm t rx = rx − rx = −24 m − 12 m = −36 m . 026 (part 2 of 3) 3.333 points Now find by . Correct answer: 7.2. Explanation: This is the same as in the first part: Find 029 (part 2 of 4) 2.5 points Answer in m. tm ry . Correct answer: 1 m. by = f × a y = −4 × −1.8 = 7. 2 . 027 (part 3 of 3) 3.333 points Finally, find bz . 6 Explanation: Same process but with the y components: tm t m ry = ry − ry = 33 m − 32 m = 1 m. schopper (as55864) – Ch1-HW1 – Antoniewicz – (56465) 030 (part 3 of 4) 2.5 points tm Find rz . Answer in m. 7 21 shown in the diagram. Start by finding rx . Answer in m. Correct answer: 30 m. Explanation: Same process but with the z components: r2 tm t m rz = rz − rz = 43 m − 13 m = 30 m . 031 (part 4 of 4) 2.5 points What is the distance from the man’s head to the top of the tree? Answer in m. Correct answer: 46.8722 m. Explanation: This distance is just the length of the vector tm r , which we can find with the Pythagorean theorem: r tm = = t m t m t m (rx − rx )2 +(ry − ry )2 +(rz − rz )2 r1 r 21 Correct answer: 10 m. Explanation: 21 To find rx , we just subtract the x components, starting with the coordinate from r 2 : 21 2 1 rx = rx − rx = 13 m − 3 m = 10 m . (−36 m)2 + (1 m)2 + (30 m)2 √ = 2197 m2 = 46.8722 m . 033 (part 2 of 3) 3.333 points 21 Find ry . Answer in m. Correct answer: 6 m. 032 (part 1 of 3) 3.333 points In the following figure, the position of object 1 is given by Explanation: Same procedure as in part 1, but with the y components: r 1 = 3 m, −3 m, 0 . The position of object 2 is given by r 2 = 1 3 m, 3 m, 0 . You will calculate the components of the relative position vector giving the position of object 2 relative to object 1. Before putting in your answers, see whether they are consistent with the appearance of the vector r 2 relative to 1 = r 21 = r 2 − r 1 21 2 1 ry = ry − ry = 3 m − (−3 m) = 6 m. 034 (part 3 of 3) 3.333 points Which of the following choices represents the position of object 1 relative to object 2? 1. r 12 = 10 m, 6 m, 0 schopper (as55864) – Ch1-HW1 – Antoniewicz – (56465) at 45◦ angles and using trigonometry. For instance, E ’s horizontal component seems to be about 3.5 units. If we call E ’s length L, then we can write down the following equation: 2. r 12 = 6 m, 10 m, 0 3. r 12 = 0, −10 m, −6 m 4. r 12 = −10 m, −6 m, 0 correct 5. r 12 = −6 m, −10 m, 0 L cos 45◦ = 3.5 ⇒L= Explanation: The relative position of object 1 to object 2 is just the negative of the vector we found in parts 1 and 2. We flip the sign of each component to find that r 12 = −10 m, −6 m, 0 . 035 (part 1 of 2) 5.0 points The following figure shows several arrows representing vectors in the xy plane. B C E D 8 A F G Which vectors have magnitudes equal to the magnitude of A? List all that apply, separated by commas. (Note: for the purposes of this problem, you may assume that if two vectors appear to be nearly the same length, they are exactly the same.) Correct answer: B, C, D, E, F. Explanation: Magnitude refers to the length of the vector. We can easily count the number of units along A to find that its length is 5. Vectors B , D, and F are also clearly 5 units, so we know those are correct choices. G is clearly not 5 units, so that one is incorrect. The remaining two vectors, C and E , are also five units long. We can determine this by noting that they are 3. 5 ≈ 5. cos 45◦ So C and E are also correct choices. 036 (part 2 of 2) 5.0 points Which vectors are equal to A? List all that apply, separated by commas. Correct answer: B, F. Explanation: Actually being equal to A requires not only equal magnitude, but that the vectors be pointing in the same direction. The only vectors that meet both of these criteria are B and F . ...
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