Use standard gravity, a = 9.80665 m/s2, for equations involving the Earth's gravitational force as the acceleration rate of an object. Calculus can be used to calculate the position, velocity, and acceleration of the asteroid at any given time, which can be used to predict its path and potential impact on Earth. This problem presents the first derivatives of the x and y coordinate positions of a particle moving along a curve along with the position of the particle at a specific time, and asks for: the slope of a tangent line at a specific time, the speed, and the acceleration vector of the particle at that time as well as the y-coordinate of the particle at another time, and the total distance traveled by the particle over a time interval. The first one relies on the basic velocity definition that uses the well-known velocity equation. Click Agree and Proceed to accept cookies and enter the site. Since we want to intercept the enemy missile, we set the position vectors equal to each other. Then, we'd just solve the equation like this: ds/dt = -3t + 10. ds/dt = -3 (5) + 10. Lets first compute the dot product and cross product that well need for the formulas. A particle's position on the-axisis given by the functionfrom. If an object's velocity is 40 miles per hour and the object accelerates 10 miles per hour per hour, the object is speeding up. Then take an online Calculus course at StraighterLine for college credit. s = 160 m + 320 m This is meant to to help students connect the three conceptually to help solidify ideas of what the derivative (and second derivative) means. 2: Vector-Valued Functions and Motion in Space, { "2.1:_Vector_Valued_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.2:_Arc_Length_in_Space" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.3:_Curvature_and_Normal_Vectors_of_a_Curve" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.4:_The_Unit_Tangent_and_the_Unit_Normal_Vectors" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.5:_Velocity_and_Acceleration" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.6:_Tangential_and_Normal_Components_of_Acceleration" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.7:_Parametric_Surfaces" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "1:_Vector_Basics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2:_Vector-Valued_Functions_and_Motion_in_Space" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3:_Multiple_Integrals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4:_Integration_in_Vector_Fields" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "acceleration vector", "projectiles", "velocity", "speed", "showtoc:no" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FCalculus%2FSupplemental_Modules_(Calculus)%2FVector_Calculus%2F2%253A_Vector-Valued_Functions_and_Motion_in_Space%2F2.5%253A_Velocity_and_Acceleration, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 2.4: The Unit Tangent and the Unit Normal Vectors, 2.6: Tangential and Normal Components of Acceleration. In order to find the first derivative of the function, Because the derivative of the exponential function is the exponential function itself, we get, And differentiatingwe use the power rule which states, To solve for the second derivative we set. To find the acceleration of the particle, we must take the first derivative of the velocity function: The derivative was found using the following rule: Now, we evaluate the acceleration function at the given point: Calculate Position, Velocity, And Acceleration, SSAT Courses & Classes in San Francisco-Bay Area. In this example, the change in velocity is determined to be 4 (m/s). Suppose that the vector function of the motion of the particle is given by $\mathbf{r}(t)=(r_1,r_2,r_3)$. In single variable calculus the velocity is defined as the derivative of the position function. If this function gives the position, the first derivative will give its speed. In one variable calculus, speed was the absolute value of the velocity. question. . x = x0 +v0t+ 1 2mv2 x = x 0 + v 0 t + 1 2 m v 2. Acceleration is negative when velocity is decreasing9. Intervals when velocity is increasing or decreasing23. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. (b) What is the position function? Get hundreds of video lessons that show how to graph parent functions and transformations. Students begin in cell #1, work the problem, and then search for their answer. The two most commonly used graphs of motion are velocity (distance v. time) and acceleration (velocity v. time). Get hundreds of video lessons that show how to graph parent functions and transformations. Since the time derivative of the velocity function is acceleration, d dtv(t) = a(t), we can take the indefinite integral of both sides, finding d dtv(t)dt = a(t)dt + C1, where C 1 is a constant of integration. Particle motion describes the physics of an object (a point) that moves along a line; usually horizontal. \], \[\textbf{v} (\dfrac{p}{4}) = 2 \hat{\textbf{j}} - \dfrac{ \sqrt{2} }{2}. Velocity is the derivative of position: Acceleration is the derivative of velocity: The position and velocity are related by the Fundamental Theorem of Calculus: where The quantity is called a displacement. Set the position, velocity, or acceleration and let the simulation move the man for you. Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. How to tell if a particle is moving to the right, left, at rest, or changing direction using the velocity function v(t)6. This means we use the chain rule, to find the derivative. The particle is moving to the right when the velocity is positive17. where \(\kappa \) is the curvature for the position function. Find the functional form of velocity versus time given the acceleration function. Next, we also need a couple of magnitudes. These equations model the position and velocity of any object with constant acceleration. There really isnt much to do here other than plug into the formulas. Find the functional form of position versus time given the velocity function. The Fundamental Theorem of Calculus says that Similarly, the difference between the position at time and the position at time is determined by the equation In this section we need to take a look at the velocity and acceleration of a moving object. The slope about the line on these graphs lives equal to the quickening is the object. To find the second derivative we differentiate again and use the product rule which states, whereis real number such that, find the acceleration function. Let \(r(t)\) be a differentiable vector valued function representing the position vector of a particle at time \(t\). Hence the particle does not change direction on the given interval. In the tangential component, \(v\), may be messy and computing the derivative may be unpleasant. Please revise your search criteria. If you're seeing this message, it means we're having trouble loading external resources on our website. 1. This helps us improve the way TI sites work (for example, by making it easier for you to find information on the site). Our anti-missile-missile starts out at base, so the initial position is the origin. \], Find the velocity vector \(\textbf{v}(t)\) if the position vector is, \[\textbf{r} (t) = 3t \hat{\textbf{i}} + 2t^2 \hat{\textbf{j}} + \sin (t) \hat{\textbf{k}} . In Instantaneous Velocity and Speed and Average and Instantaneous Acceleration we introduced the kinematic functions of velocity and acceleration using the derivative. Velocity is nothing but rate of change of the objects position as a function of time. s = ut + at2 Acceleration is zero at constant velocity or constant speed10. s = ut + at2 \], \[ 100000 \sin q = 3000 + 50000 \cos q + 15000 .\], At this point we use a calculator to solve for \(q\) to, Larry Green (Lake Tahoe Community College). Find the instantaneous velocity at any time t. b. It works in three different ways, based on: Difference between velocities at two distinct points in time. The calculator can be used to solve for s, u, a or t. Displacement (s) of an object equals, velocity (u) times time (t), plus times acceleration (a) times time squared (t2). Here is the answer broken down: a. position: s (2) gives the platypus's position at t = 2 ; that's. or 4 feet, from the back of the boat. Investigating the relationship between position, speed, and acceleration. Position-Velocity-Acceleration AP Calculus A collection of test-prep resources Help students score on the AP Calculus exam with solutions from Texas Instruments. From Calculus I we know that given the position function of an object that the velocity of the object is the first derivative of the position function and the acceleration of the object is the second derivative of the position function. Move the little man back and forth with the mouse and plot his motion. In this case, code is probably more illuminating as to the benefits/limitations of the technique. Learn about the math and science behind what students are into, from art to fashion and more. You can fire your anti-missile at 100 meters per second. \], \[ \textbf{v}_e (t)= v_1 \hat{\textbf{i}} + (v_2-9.8t) \hat{\textbf{j}} .\], Setting \(t = 0\) and using the initial velocity of the enemy missile gives, \[ \textbf{v}_e (t)= -30 \hat{\textbf{i}} + (3-9.8t) \hat{\textbf{j}}. Read More Substituting this expression into Equation \ref{3.19} gives, \[x(t) = \int (v_{0} + at) dt + C_{2} \ldotp\], \[x(t) = v_{0} t + \frac{1}{2} at^{2} + C_{2} \ldotp\], so, C2 = x0. The tangential component of the acceleration is then. Legal. Accessibility StatementFor more information contact us atinfo@libretexts.org. Vectors - Magnitude \u0026 direction - displacement, velocity and acceleration12. a = acceleration Then the velocity vector is the derivative of the position vector. This section assumes you have enough background in calculus to be familiar with integration. A particle starts from rest and has an acceleration function \(a(t)=\left(5-\left(10 \frac{1}{s}\right) t\right) \frac{m}{s^{2}}\). We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Particle motion along a coordinate axis (rectilinear motion): Given the velocities and initial positions of two particles moving along the x-axis, this problem asks for positions of the particles and directions of movement of the particles at a later time, as well as calculations of the acceleration of one particle and total distance traveled by the other. Examine the technology solutions to the 2021 AP Calculus FRQ AB2, even if the question is not calculator active. Given: y=1.0+25t5.0t2 Find: a . This video illustrates how you can use the trace function of the TI-Nspire CX graphing calculator in parametric mode to visualize particle motion along a horizontal line. Speeding Up or Slowing Down If the velocity and acceleration have the same sign (both positive or both negative), then speed is increasing. s = 160 m + 0.5 * 640 m a = acceleration To introduce this concept to secondary mathematics students, you could begin by explaining the basic principles of calculus, including derivatives and integrals. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Step 1: Enter the values of initial displacement, initial velocity, time and average acceleration below which you want to find the final displacement. Examine the technology solutions to the 2021 AP Calculus FRQ AB2, even if the question is not calculator active. What are the 3 formulas for acceleration? Additional examples are presented based on the information given in the free-response question for instructional use and in preparing for the AP Calculus . (b) At what time does the velocity reach zero? Where: Assume that gravity is the only force acting on the projectiles. The calculator can be used to solve for s, u, a or t. This helps us improve the way TI sites work (for example, by making it easier for you to find information on the site). Additional examples are presented based on the information given in the free-response question for instructional use and in preparing for the AP Calculus exam. If you do not allow these cookies, some or all site features and services may not function properly. \[\textbf{a}(t) = \textbf{v}'(t) = 2 \hat{\textbf{j}} . We can find the acceleration functionfrom the velocity function by taking the derivative: as the composition of the following functions, so that. Mathematical formula, the velocity equation will be velocity = distance / time Initial Velocity v 0 = v at Final Velocity v = v 0 + at Acceleration a = v v 0 /t Time t = v v 0 /a Where, v = Velocity, v 0 = Initial Velocity a = Acceleration, t = Time. s = 100 m + 0.5 * 48 m The tangential component is the part of the acceleration that is tangential to the curve and the normal component is the part of the acceleration that is normal (or orthogonal) to the curve. Activities for the topic at the grade level you selected are not available. (c) When is the velocity zero? With a(t) = a, a constant, and doing the integration in Equation \ref{3.18}, we find, \[v(t) = \int a dt + C_{1} = at + C_{1} \ldotp\], If the initial velocity is v(0) = v0, then, which is Equation 3.5.12. vi = initial velocity \], \[ \textbf{r} (t) = 3 \hat{\textbf{i}}+ 2 \hat{\textbf{j}} + \cos t \hat{\textbf{k}} .\]. Scalar Quantities - Speed and Distance13. We take t = 0 to be the time when the boat starts to decelerate. It is particularly about Tangential and Normal Components of Acceleration. s = 25 m/s * 4 s + * 3 m/s2 * (4 s)2 Similarly, the time derivative of the position function is the velocity function, Thus, we can use the same mathematical manipulations we just used and find, \[x(t) = \int v(t) dt + C_{2}, \label{3.19}\]. Acceleration is positive when velocity is increasing8. Example 3.2: The position of a ball tossed upward is given by the equation y=1.0+25t5.0t2. There are two formulas to use here for each component of the acceleration and while the second formula may seem overly complicated it is often the easier of the two. Calculate the position of the person at the end time 6s if the initial velocity of the person is 4m/s and angular acceleration is 3 m/s2. What is its acceleration at ? Position is the location of object and is given as a function of time s (t) or x (t). This is done by finding the velocity function, setting it equal to, and solving for. Conclusion zThe velocity function is found by taking the derivative of the position function. Notice that the velocity and acceleration are also going to be vectors as well. The mass of an accelerating object and the force that acts on it. In the study of the motion of objects the acceleration is often broken up into a tangential component, \({a_T}\), and a normal component, \({a_N}\). Particle motion in the coordinate plane: Given the vector-valued velocity and initial position of a particle moving in the coordinate plane, this problem asks for calculations of speed and the acceleration vector at a given time, the total distance traveled over a given time interval, and the coordinates of the particle when it reaches its leftmost position. The graph of velocity is a curve while the graph of acceleration is linear. When we think of speed, we think of how fast we are going. In one variable calculus, we defined the acceleration of a particle as the second derivative of the position function. s = displacement Since \(\int \frac{d}{dt} v(t) dt = v(t)\), the velocity is given by, \[v(t) = \int a(t) dt + C_{1} \ldotp \label{3.18}\]. Typically, the kinematic formulas are written as the given four equations. Let \(\textbf{r}(t)\) be a twice differentiable vector valued function representing the position vector of a particle at time \(t\). When is the particle at rest? Slope of the secant line vs Slope of the tangent line4. Calculate the radius of curvature (p), During the curvilinear motion of a material point, the magnitudes of the position, velocity and acceleration vectors and their lines with the +x axis are respectively given for a time t. Calculate the radius of curvature (p), angular velocity (w) and angular acceleration (a) of the particle for this . The TI in Focus program supports teachers in Well first get the velocity. 2006 - 2023 CalculatorSoup . To do this all (well almost all) we need to do is integrate the acceleration. 4.2 Position, Velocity, and Acceleration Calculus 1. prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x). Includes full solutions and score reporting. This is the third equation of motion.Once again, the symbol s 0 [ess nought] is the initial position and s is the position some time t later. Since the time derivative of the velocity function is acceleration, we can take the indefinite integral of both sides, finding, \[\int \frac{d}{dt} v(t) dt = \int a(t) dt + C_{1},\], where C1 is a constant of integration. Average velocity is displacement divided by time15. TI websites use cookies to optimize site functionality and improve your experience. If you do not allow these cookies, some or all of the site features and services may not function properly. How to find the intervals when the particle is moving to the right, left, or is at rest22. To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Using Derivatives to Find Acceleration - How to Calculus Tips. Velocities are presented in tabular and algebraic forms with questions about rectilinear motion (position, velocity and acceleration). when \(t = -1\). If this function gives the position, the first derivative will give its speed and the second derivative will give its acceleration. The axis is thus always labeled t (s). For example, if we want to find the instantaneous velocity at t = 5, we would just substitute "5" for t in the derivative ds/dt = -3 + 10. In this case, the final position is found to be 400 (m). Given the position function, find the velocity and acceleration functions: Here is another: Notice how we need at least an x 2 to have a value for acceleration; if acceleration is 0, then the object in question is moving at a constant velocity. In order to solve for the first and second derivatives, we must use the chain rule. \]. Position, Velocity, Acceleration. It doesn't change direction within the given bounds, To find when the particle changes direction, we need to find the critical values of. Derive the kinematic equations for constant acceleration using integral calculus. We haveand, so we have. If an object's velocity is 40 miles per hour and the object accelerates 10 miles per hour per hour, the object is speeding up. To find out more or to change your preferences, see our cookie policy page. This Displacement Calculator finds the distance traveled or displacement (s) of an object using its initial velocity (u), acceleration (a), and time (t) traveled. Substituting back into the equation for x(t), we finally have, \[x(t) = x_{0} + v_{0} t + \frac{1}{2} at^{2} \ldotp\]. Particle Motion Along a Coordinate Line on the TI-Nspire CX Graphing Calculator. This particle motion problem includes questions about speed, position and time at which both particles are traveling in the same direction. How estimate instantaneous velocity for data tables using average velocity21. Enter the change in velocity, the initial position, and the final position into the calculator to determine the Position to Acceleration. Now, try this practical . Suppose that you are moving along the x -axis and that at time t your position is given by x(t) = t3 3t + 2. You are a anti-missile operator and have spotted a missile heading towards you at the position, \[\textbf{r}_e = 1000 \hat{\textbf{i}} + 500 \hat{\textbf{j}} \], \[ \textbf{v}_e = -30 \hat{\textbf{i}} + 3 \hat{\textbf{j}} . \], Now integrate again to find the position function, \[ \textbf{r}_e (t)= (-30t+r_1) \hat{\textbf{i}} + (-4.9t^2+3t+r_2) \hat{\textbf{j}} .\], Again setting \(t = 0\) and using the initial conditions gives, \[ \textbf{r}_e (t)= (-30t+1000) \hat{\textbf{i}} + (-4.9t^2+3t+500) \hat{\textbf{j}}. The examples included emphasize the use of technology, AP Calculus-type questions, and some are left open for exploration and discussion. downloads and learning objectives related to each free-response However, our given interval is, which does not contain. of files covers free-response questions (FRQ) from past exams If you are moving along the x -axis and your position at time t is x(t), then your velocity at time t is v(t) = x (t) and your acceleration at time t is a(t) = v (t) = x (t). One method for describing the motion of an objects is through the use of velocity-time graphs which show the velocity of the obj as a function out time. Since velocity represents a change in position over time, then acceleration would be the second derivative of position with respect to time: a (t) = x (t) Acceleration is the second derivative of the position function. All the constants are zero. We will find the position function by integrating the velocity function. These cookies enable interest-based advertising on TI sites and third-party websites using information you make available to us when you interact with our sites. Using the integral calculus, we can calculate the velocity function from the acceleration function, and the position function from the velocity function. \], \[\textbf{b}(-1)= 2 \hat{\textbf{i}} - \hat{\textbf{j}} .\]. A motorboat is traveling at a constant velocity of 5.0 m/s when it starts to decelerate to arrive at the dock. Cite this content, page or calculator as: Furey, Edward "Displacement Calculator s = ut + (1/2)at^2" at https://www.calculatorsoup.com/calculators/physics/displacement_v_a_t.php from CalculatorSoup, The three acceleration formulas: a = v/t a = F/m a = 2 (d-Vit)/t How do you find acceleration with force and mass on a calculator? Use the integral formulation of the kinematic equations in analyzing motion. The TI in Focus program supports teachers in preparing students for the AP Calculus AB and BC test. This page titled 3.8: Finding Velocity and Displacement from Acceleration is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Given a table of velocity values for a particle moving along a vertical line, students calculate or approximate associated derivative and integral values, interpreting them in the context of the problem (for example; position, acceleration, etc.). Working with a table of velocity values: 2021 AP Calculus AB2 Technology Solutions and Extensions. There are 3 different functions that model this motion. (The bar over the a means average acceleration.) 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