Chapter 23. Linear Motion Motion of a Bug
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1 Chapter 23 Linear Motion The simplest example of a parametrized curve arises when studying the motion of an object along a straight line in the plane We will start by studying this kind of motion when the starting and ending locations are known 231 Motion of a Bug Example 2311 A bug is spotted at in the - plane The bug walks in a straight line from to PSfrag replacements at a constant speed It takes the bug 5 seconds to reach Assume the units of our coordinate system are feet What is the speed of the bug along the line connecting and? Compute the horizontal and vertical speeds of the bug and show they are both constant Solution A standard technique in motion problems is to analyze the and -motion separately This means we look at the projection of the bug location onto the and separately, studying how each projection moves We can think of these projections as shadows cast by a flashlight onto the two axes: PSfrag replacements For the -motion, we study the shadow on the which starts at 2 and moves toward 6 on the For the -motion, we study the shadow on the which starts at 5 and moves toward 3 along the In general, speed is computed by dividing distance by time elapsed, so dist!",- feet sec ft sec %& ')(* %+,' ft sec (231) This is the speed of the bug along the line connecting and 315 P=(2,5) Q=(6,3) (a) A bug walking from to shadow on y axis P=(2,5) light on bug casts shadow for x motion R=(x,y ) 1 1 light on bug casts shadow for y motion S=(x,y ) 2 2 Q=(6,3) 6 shadow on x axis (b) How to view motion in the coordinates Figure 231: Visualizing the model for the bug problem
2 ' 316 CHAPTER 23 LINEAR MOTION The hard part of this problem is to show that the speed of the horizontal and vertical shadows are also constant This might seem obvious when you first think about it In order to actually show it, let s take two intermediary positions and & ' ' along the bugs path We are going to relate the horizontal speed of the bug between and ', the vertical speed of the bug between and ' and the speed of the bug from to Actually, because there are positive or negative directions for the and axes, we will allow horizontal and vertical speed to be a quantity, with the obvious meaning If it takes seconds for the bug to travel from to, then is the elapsed time for the horizontal motion from to ' and also the elapsed time for the vertical motion from to ' The horizontal speed is the directed distance ' divided by the time elapsed, whereas the vertical speed is the directed distance ' divided by the time elapsed We want to show that and are both constants! To do this, we have these three equations: ' ' distance ) ' ( ' ' ' ( ' ' Now, square each side of the three equations and combine them to conclude: ' ' ' ' ( ' ' We can multiply through by ' and that gives us the key equation: ' ' ( ' On the other hand, the ratio of the vertical and horizontal speed gives ' ' slope of line connecting and Solving for in terms of we can write (232) (233) Since is positive, (234) ' ( ' ' (! " - % - '&
3 ' 232 GENERAL SETUP 317 that is, - -,- feet sec By (234), ft This shows the speed in both the horizontal and vertical directions is constant as the bug moves from to sec Since and were any two intermediary points between and, the horizontal and vertical bug speeds are constant Example 2312 Parametrize the motion in the previous problem Solution We have shown feet/sec and feet/sec, so -coordinate of the bug at time beginning -coordinate( the -direction in seconds ( -coordinate of the bug at time beginning -coordinate( the -direction in seconds As a check, notice that, ( %*,) is the starting location and ( is the ending location 232 General Setup Given two points and ' ' in the plane, we can study motion of an object along thepsfrag line connecting replacements and In so doing, you need to first specify the starting location and the ending location of the object; lets say we start at and proceed to Fix the distance units used in the coordinate system (feet, inches, miles, meters, etc) and the time units used (seconds, hours, years, etc) As highlighted in the solution of Example 2312, the key is to analyze the -motion and -motion separately starting location P=(x,y ) 1 1 We will be imposing an assumption that the speed along the line connecting and in Figure 231(b) is a constant There is a crucial observation we need to make, a special case of which was the content of Example 2311 Important Facts 2321 (Linear motion) Assume that an object moves from to along a straight line at a constant speed, as in Figure 232 ending location Q=(x,y ) 2 2 moving from P to Q at a constant speed Figure 232: Typical scenario
4 318 CHAPTER 23 LINEAR MOTION ments!!! CAUTION!!! Then the speed in the -direction and the speed in the -direction are both constant We also have two useful formulas: ' ' ( ' slope of line of travel, when the line is nonvertical This fact is established using the same reasoning as in Example 2311 Let s make a few comments To begin with, if the line of travel is either vertical or horizontal, then either or and Fact 2321 isn t really saying anything of interest The formulas in Fact 2321 only work for linear motion Q P Q Sfrag replacements no y motion P no x motion Figure 233: Horizontal or vertical motion For any other line of travel, we can use the reasoning used in Example 2311 Pay attention that the horizontal speed and the vertical speed are both directed quantities; ie these can be positive or negative The sign of will indicate the direction of motion: If is positive, then the horizontal motion is to the right and if is negative, then the horizontal motion is to the left Similarly, the sign of tells us if the vertical motion is upward or downward Returning to Figure 232, to describe the -motion, two pieces of information are needed: the starting location (in the -direction) and the constant speed in the -direction So, -coordinate of the object at time beginning -coordinate( ( the -direction in time units
5 232 GENERAL SETUP 319 If we are not given the horizontal velocity directly, rather the time required to travel from to, then we could compute using the fact that the object starts at " and travels to " ' : directed horizontal distance traveled time required to travel this distance ending -coordinate starting -coordinate time required to travel this distance ' To describe the -motion in Figure 232, we proceed similarly We will denote by the constant vertical speed of the object, then after time units the object has traveled units So, -coordinate of object at time In summary, beginning -coordinate( ( the -direction in time units Important Fact 2322 (Linear motion) Suppose an object begins at a point and moves at a constant speed along a line connecting to another point ' ' Then the motion of this object will trace out a line segment which is parametrized by the equations: ( + ( Example 2323 Return to the linear motion PSfrag problem replacements studied in Example 2311 and 2312 However, now assume that the point * is located at the center of a circular region of radius 1 ft When and where does the bug enter this circular region? P=(2,5) Q=(6,3) location E where bug enters region circular region radius 1 ft Solution The parametric equations for the linear motion of the bug are given by: Figure 234: A bug crosses a circular boundary " ( + The equation of the boundary of the circular region centered at by, ' ( & ' is given
6 320 CHAPTER 23 LINEAR MOTION To find where and when the bug crosses into the circular region, we determine where and when the linear path and the circle equation have a simultaneous solution To find such a location, we simply plug " and into the circle equation:, ' ( ' %(, ' ( ' ( ' ( ( ' ' ( Notice, this is an equation in the single variable Finding the solution of this equation will tell us when the bug crosses into the region Once we know when the bug crosses into the region, we can determine the location by plugging this time value into our parametric equations By the quadratic formula, we find the solutions are or, We know that the bug reaches the point in 5 seconds, so the second solution seconds is the time when the bug crosses into the circular region (If the bug had continued walking in a straight line directly through, then the time when the bug leaves the circular region would correspond to the other solution seconds) Finally, the location of the bug when it crosses into the region is
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