Supplementary Problems 1.18 Starting from the center of town, a car travels east for \( 80.0 \mathrm{~km} \) and then turns due south for another \( 192 \mathrm{~km} \), at which point it runs out of gas. Determine the displacement of the stopped car from the center of town. Ans. \( 208 \mathrm{~km}-67.4^{\circ} \) SOUTH OF EAST 1.19 A little turtle is placed at the origin of an \( x y \)-grid drawn on a large sheet of paper. Each grid box is \( 1.0 \mathrm{~cm} \) by \( 1.0 \mathrm{~cm} \). The turtle walks around for a while and finally ends up at point \( (24,10) \), that is, 24 boxes along the \( x \)-axis, and 10 boxes along the \( y \)-axis. Determine the displacement of the turtle from the origin at the point. Ans. \( 26 \mathrm{~cm}-23^{\circ} \) ABOVE \( X \)-AXIS 1.20 A bug starts at point \( A \), crawls \( 8.0 \mathrm{~cm} \) east, then \( 5.0 \mathrm{~cm} \) south, \( 3.0 \mathrm{~cm} \) west, and \( 4.0 \mathrm{~cm} \) north to point \( B \). (a) How far north and east is \( B \) from \( A \) ? (b) Find the displacement from \( A \) to \( B \) both graphically and algebraically. Ans. (a) \( 5.0 \mathrm{~cm} \)-EAST, \( 1.0 \mathrm{~cm} \)-NORTH; (b) \( 5.10 \mathrm{~cm}-11.3^{\circ} \) SOUTH OF EAST 1.21 Find the scalar \( x \) - and \( y \)-components of the following displacements in the \( x y \)-plane: (a) \( 300 \mathrm{~cm} \) at \( 127^{\circ} \) and (b) \( 500 \mathrm{~cm} \) at \( 220^{\circ} \). Ans. (a) \( -180 \mathrm{~cm}, 240 \mathrm{~cm} ;(b)-383 \mathrm{~cm},-321 \mathrm{~cm} \)
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Question 1., parts a. and b. For each part: All work must be shown for credit, including your correctly labeled triangle for part a. A ship sailing due east in the North Atlantic has been ordered to immediately change course because the next 50 miles of its (due east) course contains dangerous icebergs. The captain turns and sails on a bearing of 62⁰, then changes course again to a bearing of 115⁰ until the ship reaches its original course. a. How much farther did the ship have to travel to avoid the icebergs? Additional Distance = _______ miles b. Find the area of the triangle created by the new path the ship sailed to avoid the icebergs and the original (due east) path. Area = _______ square miles Question 2. All work must be shown for credit, including your correctly labeled triangle. Suppose, at the moment the ship changed course to a bearing of 115⁰, one of the icebergs (on the original due east course) was now directly south of the ship. How far away was that iceberg from the ship when it made this change in course? Distance = _______ miles Question 3. All work must be shown for credit, including your correctly labeled triangle. Now, let’s change the original question. Suppose the icebergs covered only 30 (not 50) miles. Also, when (a) the ship turned left to avoid the icebergs, suppose it traveled 15 miles. After this, suppose (b) it then turned right and traveled 18 miles to get back onto its original due east course. What were the bearings the captain had to use at (a) and (b) above in order to sail on this new path? IMPORTANT: This means that, after you have drawn the triangle and determined its angles, you are not finished! You need to actually find each bearing (clockwise angle measured from due north): (a) The bearing just after the ship turned left to avoid the icebergs, and (b) The bearing just after it turned right in order to travel back to its original course. Bearing at (a) = _______ ⁰ Bearing at (b) = _______ ⁰
Michelle M.
As she picks up her riders, a bus driver traverses four successive displacements represented by the expression $$\begin{array}{l} (-6.30 \mathrm{b}) \hat{\mathrm{i}}-\left(4.00 \mathrm{b} \cos 40^{\circ}\right) \hat{\mathrm{i}}-\left(4.00 \mathrm{b} \sin 40^{\circ}\right) \hat{\mathrm{j}} \\ +\left(3.00 \mathrm{b} \cos 50^{\circ}\right) \hat{\mathrm{i}}-\left(3.00 \mathrm{b} \sin 50^{\circ}\right) \hat{\mathrm{j}}-(5.00 \mathrm{b}) \hat{\mathrm{j}} \end{array}$$ Here b represents one city block, a convenient unit of distance of uniform size; i is east; and $\hat{\mathbf{j}}$ is north. The displacements at $40^{\circ}$ and $50^{\circ}$ represent travel on roadways in the city that are at these angles to the main east-west and northsouth streets. (a) Draw a map of the successive displacements. (b) What total distance did she travel? (c) Compute the magnitude and direction of her total displacement. The logical structure of this problem and of several problems in later chapters was suggested by Alan Van Heuvelen and David Maloney, American Journal of Physics $67(3) 252-256$ March 1999.
Cars $A$ and $B$ move in the same direction in adjacent lanes. The position $x$ of car $A$ is given in Fig. $2-30,$ from time $t=0$ to $t=7.0 \mathrm{~s}$. The figurc's vertical scaling is sct by $x_{x}=32.0 \mathrm{~m} .$ At $t=0,$ car $B$ is at $x=0,$ with a velocity of $12 \mathrm{~m} / \mathrm{s}$ and a negative constant acceleration $a_{B-}$ (a) What must $a_{0}$ be such that the cars are (momentarily) side by side (momentarily at the same value of $x$ ) at $t=4.0 \mathrm{~s} ?$ (b) For that value of $a_{n}$, how many times are the cars side by side? (c) Sketch the position $x$ of car $B$ versus time $t$ on Fig. 2 - 30 . How many times will the cars be side by side if the magnitude of acceleration $a_{B}$ is (d) more than and (e) less than the answer to part (a)?
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