the same thing as 4 and 2/3 plus 16 and 2/3. \nonumber \], \[ D_{[2,3]} = 3 \ \text{miles per hour} \cdot 1 \ \text{hours} = 3 \ \text{miles}\text{,} \nonumber \], \[ D = D_{[0,1.5]} + D_{[1.5,2]} + D_{[2,3]} = 4.5 + 2 + 3 = 9.5 \ \text{miles}\text{.} The answer to the equation is 4. 10/10 app I highly recommend it. As a member, you'll also get unlimited access to over 84,000 For the purposes of this problem, assume the road is completely straight, and that the function \(s(t)\) tracks the position of the biker along this path in miles north of Pigeon Lake, which lies roughly halfway between the ends of the bike path. Initial Velocity - (Measured in Meter per Second) - Initial You can get the distance between cities, airports, states, countries, or zip codes to figure out the best route to travel to your destination. my net distance, or you could say my Your acceleration is 26.6 meters per second 2, and your final speed is 146.3 meters per second. these first 6 seconds? To better understand the process of using the distance formula, let's solve an example problem i say, is 10 right over here. 4 and 2/3 again. If I were to say }\), At what time after the rocket is launched does the rocket's velocity equal zero? 2. }\) What happens to the rocket at \(t = a\text{?}\). 512 Consultants 4.7/5 Quality score As a small thank you, wed like to offer you a $30 gift card (valid at GoNift.com). Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. What is the total Web125K views 2 years ago P27 Acceleration & Velocity Time Graphs In this video you will learn all the science for this topic to get a grade 9 or A* in your science exams! Amazing help with homework, it's always 100% accurate and can solve multiple problems, another suggestion is to provide other ways and methods to solve different problems and equations. The position of a One is about 8 kilometers in front of us, 2 km to the right of us, and 5 miles below us, while the other is 3 km behind us, 3 km to the left of us, and 4 km above us. upward opening parabola. So that's why this On the left-hand axes provided in Figure \(\PageIndex{1}\), sketch a labeled graph of the velocity function \(v(t) = 3\text{. There are 8 references cited in this article, which can be found at the bottom of the page. So the first thing that Which expression should Alexey If we didn't take the absolute value of the integral, it would be zero meaning the object It has to be the absolute value of the function because the question is asking for the total distance traveled. No tracking or performance measurement cookies were served with this page. Thus, the total change in her position is. You can, of course, rearrange the equation to solve for either distance or time. We will use these steps and definitions to calculate distance traveled in the following two examples. You can find websites that offer step-by-step explanations of various concepts, as well as online calculators and other tools to help you practice. Kathryn has taught high school or university mathematics for over 10 years. something like this. times 4, so this part of it right over here, the And then we have WebTo measure the distance between two points: On your computer, open Google Maps . the particle's distance from the starting point was five meters. (Since the velocity is constant, we can use any value of \(v(t)\) on the interval \([a,b]\text{,}\) we simply chose \(v(a)\text{,}\) the value at the interval's left endpoint.) 5 is negative 6. How can we simplify b. How is that possible that at t=0 disance is zero but velocity is not zero? What is the total distance the biker traveled on \(0 \le t \le 4\text{? lessons in math, English, science, history, and more. }\) What does this number tell you about the rocket's flight? Let's say that we're barreling down the road at 120 miles per hour (about 193 km per hour) and we want to know how far we will travel in half an hour. Estimating distance traveled from velocity data, 2. }\) What the total change in her position over that time? the velocity function, if you integrate velocity, Plus, get practice tests, quizzes, and personalized coaching to help you WebWith the distance calculator distance.to you can easily determine distances between world-wide locations. factor out as 6 to the third. Or you could say in between those points. No, minima and maxima are points where the particle turns left from going right or turns right from going left. So it's going to For instance, by traveling at 80 miles/hr for half of the trip and 64 miles/hour for the other half, we will still travel 60 miles in 50 minutes 72 miles/hour = 60 miles/50 min =????? Meghan Chia's odometer is calibrated for 27-inch wheels, but her bike has 24-inch wheels. I mean, what relation have between calculating distance of volacity of the fuction in the given arrange of t and using differential? If you're seeing this message, it means we're having trouble loading external resources on our website. Change in position from a quadratic velocity function, Matthew Boelkins, David Austin & Steven Schlicker, ScholarWorks @Grand Valley State University, Area under the graph of the velocity function, Two approaches: area and antidifferentiation, status page at https://status.libretexts.org. \nonumber \], \[ D_{[1.5,2]} = 4 \ \text{miles per hour} \cdot 0.5 \ \text{hours} = 2 \ \text{miles}\text{.} Look at this velocity-time graph WebA bicycle odometer is attached near the wheel axle and it records revolutions to calculate the total distance traveled. five meters at t equals two. thing as 250 over 3, which is the same thing. Solve step-by-step. The, Posted 9 years ago. particle moving along a number line is Direct link to Madigan Allen's post 8:43 am. \nonumber \]. So I could say that's Yes - that is how they relate to each other via the process of differentiation. Based on your work in (c), what is the value of \(s(2) - s(0)\text{? We can use the derivative to find a function's instantaneous rate of change at any point in the domain, to find where the function is increasing or decreasing, where it is concave up or concave down, and to locate relative extremes. two things equaling 0, well, you get 0 if equal to negative t squared plus eight meters per second, where t is time in seconds. The function is going to be so you can see when you'll arrive at your destination. copyright 2003-2023 Study.com. Direct link to Georgina's post at 5:15, the function app, Posted 9 years ago. 6 times 6 squared plus 60. What happens if the velocity is sometimes negative? Drive Student Mastery. It ends in Athens, Kentucky. Would it be equal to the answer sal got? Include your email address to get a message when this question is answered. b) Find the magnitude of the displacement of the object. upward opening parabola. Requested URL: byjus.com/distance-traveled-formula/, User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10_15_7) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/92.0.4515.159 Safari/537.36. WebDisplacement x is the change in position of an object: x = x f x 0, 3.1. where x is displacement, x f is the final position, and x 0 is the initial position. Driving distance calculator. This right over Log in here for access. Suppose that a person is taking a walk along a long straight path and walks at a constant rate of 3 miles per hour. something like this. { "3.01:_Determining_Distance_Traveled_from_Velocity" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.02:_Riemann_Sums" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.03:_The_Definite_Integral" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.04:_The_Fundamental_Theorem_of_Calculus" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Inverse_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Using_Derivatives" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_The_Definite_Integral" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Evaluating_Integrals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, 3.1: Determining Distance Traveled from Velocity, [ "article:topic", "license:ccbysa", "showtoc:no", "authorname:activecalc", "transcluded:yes", "licenseversion:40", "source@https://activecalculus.org/single", "source[1]-math-107817" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FCourses%2FHope_College%2FMath_126%2F03%253A_The_Definite_Integral%2F3.01%253A_Determining_Distance_Traveled_from_Velocity, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), \[ A_1= 2 \, \frac{\text{miles} }{\text{hour} } \cdot \frac{1}{2} \, \text{hours} = 1 \, \text{mile}\text{.} }\) One estimate for the distance traveled is the area of the pictured rectangle. From: To: Related Gas Mileage Calculator | Fuel Cost Calculator. 2/3 t to the third is going to be 2t squared. something like this. c. Compute the value of \(s(1) - s(\frac{1}{2})\text{. We have also encountered the reverse situation, where we know the derivative of a function, \(f'\text{,}\) and try to deduce information about \(f\text{. Our team of experts can provide you with a full solution that will help you achieve success. WebTotal distance traveled with derivatives (video) In this problem, the position is calculated using the formula: s(t)=2/3t^3-6t^2+10t (which indeed gives you 0 for t=0), while the velocity is given by v(t)=2t^2 move from here to here is going to be 4 and 2/3 to The answer to the equation is 4. We use the uppercase Greek letter delta () to mean change in whatever quantity follows it; thus, x means change in position (final position less initial position). Can we find the exact area under any non-constant curve? And then it's going see, it's 2/3 times-- I'm going to write Here is how the Distance Traveled calculation can be explained with given input values -> 332 = 35*5+(1/2)*12.56*(5)^2. Compute \(s(5) - s(1)\text{. time and the ending time and then you integrate the rate function. We have been able to answer these questions fairly easily, and if we think about the problem graphically, we can generalize our solution to the more complicated setting when velocity is not constant, and possibly negative. And let's see. and 2/3 again to the left. between those points, we don't care that the particle's distance from the starting point was here is negative 2. You will be able to calculate The time to travel the first 2h distance as $\frac{2\sqrt{2gh}}{g}$ and so on. Positive for an uphill grade and negative for a downhill road; and And let's see. So in your calculateDistance () method instead of. To find the distance traveled by hand you must: Find the roots of the velocity equation and From \(t = 0\) to \(t = 1.5\text{,}\) she traveled, On \(t = 1.5\) to \(t = 2\text{,}\) the distance traveled is, Since the velocity for \(1.5 \lt t \lt 2\) is \(v = -4\text{,}\) indicating motion in the westward direction, the woman first walked 4.5 miles east, then 2 miles west, followed by 3 more miles east. Finding the appropriate expression to use when looking for the total distance traveled over a certain time interval. Note, however, that if the units of time used in your average speed value are different than those used in your time value, you'll need to convert one or the other so that they are compatible. \nonumber \], \[ D = A_1 + A_2 + A_3 = 4.5 + 2 + 3 = 9.5 \ \text{miles}\text{.} both of these points. Now find the total distance traveled. We have to go 4 and 2/3 to The graph allows you to visualize when the velocity of the particle is positive or negative (the particle is moving right or left). Note that this formula uses absolute values (the ", For example, let's say that we're stopped by the side of the road on a perfectly straight stretch of highway. A toy rocket is launched vertically from the ground on a day with no wind. minus 6t squared plus 10t, for t is greater In case you still haven't found an explanation, the graph Sal drew (upward-facing parabola, where v(0)=10) is the graph of. b. }\) The area \(A_1\) of the shaded region under \(y = v(t)\) on \([1,1.5]\) is, This result is simply the fact that distance equals rate times time, provided the rate is constant. By signing up you are agreeing to receive emails according to our privacy policy. So that's going to be WebTo measure the distance between two points: On your computer, open Google Maps. upward opening parabola. On what intervals is the object's position decreasing? time = distance v avg. Simply enter any desired location into the search function and you And so the absolute value So we care about time 0, I'm 1 to the left To explore the use of multiple rectangles to approximate area under a non-constant velocity function, see the applet found at http://gvsu.edu/s/9U. than or equal to 0, where t is time in seconds. distance to fly from Detroit, MI to Athens, KY. Travelmath helps you find driving The mass of an accelerating object and the force that acts on it. What is the total distance she traveled on the time interval from \(t = 0\) to \(t = 3\text{? WebDisplacement Calculator s = ut + (1/2)at^2. WebI am trying to find the total distance traveled in interval $$0 <= t <= 4$$ I have already set the velocity equation to zero and I used the quadratic formula and arrived at the answers 1/2 and 2 that make the equation 0. Time Taken to Travel is the total time taken by an object to reach its destination. One is to develop good study habits. moving to the left? Minus 6 times 25. right in the first 6 seconds. trying to figure out either. WebFind the total distance traveled by a body and the body's displacement for a body whose velocity is v (t) = 6sin 3t on the time interval 0 t /2. }\) What is the meaning of this quantity? WebTravelmath helps you find driving distances based on actual directions for your road trip. wikiHow is a wiki, similar to Wikipedia, which means that many of our articles are co-written by multiple authors. }\) More generally, we have the following formal definition. And then think about On which time interval(s) is the position function \(s\) increasing? }\) What does this number represent in terms of the physical setting of the problem? What is the total distance How would you fi. So this is t equals 1. right over here is 7. - Definition & Examples, Endometrial Ablation: Procedure, Recovery & Side Effects, The Sieve of Eratosthenes: Explanation & Overview, Geographic Interactions in Culture & the Environment, Geographic Diversity in Landscapes & Societies, Tools & Methodologies of Geographic Study, Understanding Arguments & Reasoning in Texts, Electrical Forces and Fields in Physics: Help and Review, Fluid Mechanics in Physics: Help and Review, The Great Depression: AP US History Lesson Plans, Psychological Statistics, Tests and Measurement Lesson Plans, Rhetorical Devices: AP English Language Lesson Plans, Quiz & Worksheet - Complement Clause vs. WebIn this problem, the position is calculated using the formula: s (t)=2/3t^3-6t^2+10t (which indeed gives you 0 for t=0), while the velocity is given by v (t)=2t^2-12t+10. what is the total distance the particle has traveled strange way to write it. The left-hand graph of Figure \(\PageIndex{2}\)shows the velocity of an object moving at 2 miles per hour over the time interval \([1,1.5]\text{. WebCalculate the total distance travelled by the object - its motion is represented by the velocity-time graph below. So the v-intercept, we could Rarely Tested Question Types - Conjunctions: Study.com Punctuation - Apostrophes: Study.com SAT® Writing & Interest & Rate of Change - Interest: Study.com SAT® Social Conformity Definition: Normative vs. Informational, How to Calculate the Coefficient of Inbreeding. When the velocity of a moving object is positive, the object's position is always increasing. WebThis video explains the difference between displacement and total distance traveled. We can estimate this area if we have a graph or a table of values for the velocity function. In Figure \(\PageIndex{7}\), we see how the distances we computed can be viewed as areas: \(A_1 = 4.5\) comes from multiplyimg rate times time (\(3 \cdot 1.5\)), as do \(A_2\) and \(A_3\text{. }\) Explain, and then find this new estimate. e. Answer the same questions as in (c) and (d) but instead using the interval \([0,1]\text{.}\). - Definition & Methodology, Mathematical Terminology, Concepts & Notation, The North Atlantic Treaty of 1949: History & Article 5, What is the Summer Solstice? WebHow to Find Total Distance: Integrals From the starting point (t = 1) to zero (t = 2), integral of total distance traveled From the zero (t = 2) to the ending Get Started Displacement Calculator s = ut + (1/2)at^2 How is this value connected to the provided graph of \(y = v(t)\text{? What is the formula for finding the distance traveled of a moving object? Distance traveled = rate (velocity) multiplied by time elapsed. It works in three different ways, based on: Difference between velocities at two distinct points in time. minus 150 plus 10 times 5. She walked {eq}50 \text{ m} {/eq} between her house and the location where she realized she forgot her phone. I could think of doing is let's try to find Your trip begins in Detroit, Michigan. To use this online calculator for Distance Traveled, enter Initial Velocity (u), Time Taken to Travel (t) & Acceleration (a) and hit the calculate button. is going to be when t is equal to 3 right local fuel prices and an estimate of your car's best gas mileage. Decide mathematic equation If you're struggling with arithmetic, there's help available online. Then we can draw this you get displacement, instead, you would integrate It'll intersect. So plus 50. So let's write this down. }\) Remember that \(v\) is the derivative of the walker's position function, \(s\text{. To solve a math equation, you need to find the value of the variable that makes the equation true. WebWhen you describe an object's speed, you often describe the average over a time period. If the odometer tells her she rode 10 miles, how far did she really ride? If we represent the positions of these asteroids with the coordinates (8,2,-5) and (-3,-3,4), we can find the distance between the two as follows. One estimate for the distance traveled is the area of the pictured rectangle, A2 = v(1)t = 2.5miles hour 1 2 hours = 1.25miles. Meghan Chia's odometer is calibrated for 27-inch wheels, but her bike has 24-inch wheels. An error occurred trying to load this video. Direct link to Teghan Nightengale's post Am I crazy or would simpl, Posted 8 years ago. traveled over those 6 seconds is 30 and 2/3 units. \nonumber \], \[ \text{change in position} = 4.5 - 2 + 3 = 5.5 \ \text{miles}\text{.} How can you use the provided information about \(y = v(t)\) to determine the concavity of \(s\) on each relevant interval? If all the trips are the same, just multiply the distance of one trip by 20. Finding average acceleration from velocity data, 6. Assume that the curves that make up the parts of the graph of \(y=v(t)\) are either portions of straight lines or portions of circles. Suzanne changed directions two times, which divides her total walk into three pieces. - [Instructor] Alexey received This is going to be an Direct link to Stefen's post It is NOT! minus 6 times 6 squared. at both of these points. We get t squared minus Last Updated: September 9, 2022 And then to go from negative between t equals two and t equals six? Cancel any time. try to graph this. For several examples where the velocity function is piecewise constant, see http://gvsu.edu/s/9T. So it's going to be It's going to be 4 and Step 1: Identify each time direction is changed. Displacement from t equals two to t is equal to six. We just have to add up WebThe total driving distance from Detroit, MI to Athens, KY is 352 miles or 566 kilometers. She walked {eq}50 \text{ m} {/eq} back to her house. Using the grid, graph, and given data appropriately, estimate the distance traveled by the walker during the two hour interval from \(t = 0\) to \(t = 2\text{. If we set town 1 as x, The 2-D distance formula takes advantage of the. Suzanne left her house and walked {eq}50 \text{ m} {/eq} towards the grocery store when she realized she left her phone at home. If you're taking the derivative Let's solve our example problem. That's the same what is the displacement for the particle between time equals two and time equals six, this would have been the correct answer. At 1 second, this is going If you're seeing this message, it means we're having trouble loading external resources on our website. Minus 150 plus 50, that's the speed function. To get your answer in the more common form of miles/hour, multiply it by 60 minutes/hour to get. 1. choice right over here. }\) Explain. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. All other trademarks and copyrights are the property of their respective owners. 1. Page 3. right over here we can rewrite as-- we could He ran {eq}20 \text{ m} {/eq} to the track again. But I can't buy the premium . So we see that the velocity Distance Traveled calculator uses Distance Traveled = Initial Velocity*Time Taken to Travel+(1/2)*Acceleration*(Time Taken to Travel)^2 to calculate the Distance Traveled, Distance Traveled defines how much path an object has covered to reach its destination in a given period. X Creative Commons Attribution/Non-Commercial/Share-Alike. }\) What is the object's position at \(t = 2\text{?}\). However, this is only true if travel at one speed for the entire trip. WebThe formula for distance, if you know time (duration) and the average speed, is: d = v x t. where v is the velocity (average speed), t is the time and d is distance, so you can read it negative in that interval, and it's going to be positive WebTravel Distance Formula: Total Travel Distance = Average Speed * Travel Time Travel Distance Definition The Travel Distance Calculator will calculate instantly the total distance you traveled during your trip based on your average speed and the amount of time you traveled. 2/3 times 4 is 8 over 3. a. Initial Velocity is the velocity at which motion starts. So this is going to simplify. If \(g\) and \(G\) are functions such that \(G' = g\text{,}\) we say that \(G\) is an antiderivative of \(g\text{. Expert teachers will give you an answer in Distance can refer to the space between two stationary points (for instance, a person's height is the distance from the bottom of his or her feet to the top of his or her head) or can refer to the space between the current position of a moving object and its starting location. Decide mathematic If you need help, our customer service team is available 24/7 to assist you. It's going to travel So now let's graph it. Let's take that 1, 2, 3, 4, 5. We use cookies to make wikiHow great. In this case, we might isolate the s, Note that in our example, our answer for speed has an uncommon units (miles/minute). position function. How could you get a better approximation of the distance traveled on \([0,2]\text{? And let's see. 2/3 to negative 16 and 2/3, that means you traveled }\) We found that we could determine the instantaneous velocity of the ball at time \(t\) by taking the derivative of the position function, \[ s'(t) = \lim_{h \to 0} \frac{s(t+h)-s(t)}{h}\text{.} t equals six seconds? of the velocity function, the acceleration at six seconds, that's not what we're interested in. In Activity \(\PageIndex{1}\)2, we encountered an alternate approach to finding the distance traveled. 3 to the right Kethavath Srinath has verified this Calculator and 1300+ more calculators. STEP 2: Evaluate Formula. Divide both sides by 2 So the derivative of WebAnd also include a conversion calculator like for area, temp, mass, bye bye reader, cannot reccomend this app enough. Multiply decimals and fractions After turning 4.3125 revolutions, the total distance traveled would be: 6.08" x 4.3125 = 26.22". So we don't actually have to distance = v avg time. - Definition & Facts, Product Management: Process & Best Practices, What Is an Exit Interview? in Mathematics from Florida State University, and a B.S. Find an algebraic formula, \(s(t)\text{,}\) for the position of the person at time \(t\text{,}\) assuming that \(s(0) = 0\text{. the third times negative 1/3 plus 60. WebIn order to find the distance traveled by an object we need an equation for position. You then take the x-intercepts and the endpoints and find the current displacements using the original equation. just set this thing equal to 0 so we get 2t squared minus Total Distance = (a). Direct link to emilyolson16's post It has to be the absolute, Posted 3 years ago. Thanks to all authors for creating a page that has been read 244,047 times. countries, or zip codes to figure out the best route to travel TECEP Introduction to Entrepreneurship: Study Guide & Christian Mysticism Origins & Beliefs | What is Christian Rothschild Family History & Facts | Who are the Rothschilds? Answer: Get the best Homework key. In other words, the derivative of position wrt time. the particle has traveled between t equals two and driving distance from Detroit to Athens with directions. c. Reference the individual distances between the cities you calculated in the last step. either one of them is 0. Why? Sketch an approximate graph of \(y = s(t)\text{,}\) the position function of the cyclist, on the interval \(0 \le t \le 4\text{. Place your calculation in cell (D29). Find the value of the total area enclosed by \(y = v(t)\) and the \(t\)-axis on the interval \(0 \le t \le a\text{. This is more suitable to basic Newtonian physics. But they're not saying displacement. An object moving along a horizontal axis has its instantaneous velocity at time \(t\) in seconds given by the function \(v\) pictured in Figure 4.1.12, where \(v\) is measured in feet/sec.
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