Mathematics
Constant Acceleration Equations
Constant acceleration equations are mathematical formulas used to describe the motion of an object with a constant acceleration. These equations include the equations of motion, such as the equations for displacement, velocity, and acceleration, under constant acceleration. They are commonly used in physics and engineering to analyze the motion of objects under the influence of gravity or other forces.
Written by Perlego with AI-assistance
Related key terms
1 of 5
10 Key excerpts on "Constant Acceleration Equations"
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
It equals the change ∆ → v in the velocity divided by the elapsed time ∆t, the change in the velocity being the final minus the initial velocity; see Equation 2.4. When ∆t becomes infinitesimally small, the average acceleration becomes equal to the instantaneous acceleration → a , as Focus on Concepts 51 indicated in Equation 2.5. Acceleration is the rate at which the veloc- ity is changing. ¯ → a = ∆ → v ___ ∆t (2.4) → a = lim Δt→0 ∆ → v ___ ∆t (2.5) 2.4 Equations of Kinematics for Constant Acceleration/ 2.5 Applications of the Equations of Kinematics The equa- tions of kinematics apply when an object moves with a constant acceleration along a straight line. These equations relate the dis- placement x − x 0 , the acceleration a, the final velocity υ, the initial velocity υ 0 , and the elapsed time t − t 0 . Assuming that x 0 = 0 m at t 0 = 0 s, the equations of kinematics are as shown in Equations 2.4 and 2.7–2.9. υ = υ 0 + at (2.4) x = 1 _ 2 (υ 0 + υ)t (2.7) x = υ 0 t + 1 _ 2 at 2 (2.8) υ 2 = υ 0 2 + 2ax (2.9) 2.6 Freely Falling Bodies In free-fall motion, an object experi- ences negligible air resistance and a constant acceleration due to grav- ity. All objects at the same location above the earth have the same acceleration due to gravity. The acceleration due to gravity is directed toward the center of the earth and has a magnitude of approximately 9.80 m/s 2 near the earth’s surface. 2.7 Graphical Analysis of Velocity and Acceleration The slope of a plot of position versus time for a moving object gives the object’s velocity. The slope of a plot of velocity versus time gives the object’s acceleration. Focus on Concepts Online Time Position A B C QUESTION 8 Section 2.6 Freely Falling Bodies 6. A rocket is sitting on the launch pad. The engines ignite, and the rocket begins to rise straight upward, picking up speed as it goes. At about 1000 m above the ground the engines shut down, but the rocket continues straight upward, losing speed as it goes.
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
In terms of models, when the acceleration of a particle is zero, the particle under constant acceleration model reduces to the par- ticle under constant velocity model (Section 2.3). Equations 2.13 through 2.17 are kinematic equations that may be used to solve any problem involving a particle under constant acceleration in one dimension. These equations are listed together below for convenience. The choice of which equation you use in a given situation depends on what you know beforehand. Some- times it is necessary to use two of these equations to solve for two unknowns. You should recognize that the quantities that vary during the motion are position x f , velocity v xf , and time t. You will gain a great deal of experience in the use of these equations by solving a number of exercises and problems. Many times you will discover that more than one method can be used to obtain a solution. Remember that these equations of kinematics cannot be used in a situation in which the acceleration varies with time. They can be used only when the acceleration is constant. Q UICK QUIZ 2.7 In Figure 2.12, match each v x –t graph on the top with the a x –t graph on the bottom that best describes the motion. Velocity as a function of position for the particle under constant acceleration model t v x a t v x b t v x c t a x d t a x e t a x f Figure 2.12 (Quick Quiz 2.7) Parts (a), (b), and (c) are v x –t graphs of objects in one- dimensional motion. The possible accelerations of each object as a function of time are shown in scrambled order in (d), (e), and (f). Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
Chapter 2 | Kinematics 77 • Average speed is the total distance traveled divided by the elapsed time. (Average speed is not the magnitude of the average velocity.) Speed is a scalar quantity; it has no direction associated with it. 2.4 Acceleration • Acceleration is the rate at which velocity changes. In symbols, average acceleration a - is a - = Δv Δt = v f − v 0 t f − t 0 . • The SI unit for acceleration is m/s 2 . • Acceleration is a vector, and thus has a both a magnitude and direction. • Acceleration can be caused by either a change in the magnitude or the direction of the velocity. • Instantaneous acceleration a is the acceleration at a specific instant in time. • Deceleration is an acceleration with a direction opposite to that of the velocity. 2.5 Motion Equations for Constant Acceleration in One Dimension • To simplify calculations we take acceleration to be constant, so that a - = a at all times. • We also take initial time to be zero. • Initial position and velocity are given a subscript 0; final values have no subscript. Thus, Δt = t Δx = x − x 0 Δv = v − v 0 ⎫ ⎭ ⎬ • The following kinematic equations for motion with constant a are useful: x = x 0 + v - t v - = v 0 + v 2 v = v 0 + at x = x 0 + v 0 t + 1 2 at 2 v 2 = v 0 2 + 2a(x − x 0 ) • In vertical motion, y is substituted for x . 2.6 Problem-Solving Basics for One-Dimensional Kinematics • The six basic problem solving steps for physics are: Step 1. Examine the situation to determine which physical principles are involved. Step 2. Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Step 3. Identify exactly what needs to be determined in the problem (identify the unknowns). Step 4. Find an equation or set of equations that can help you solve the problem. Step 5. Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
However, we can derive other equations that might prove useful in certain specific situations. First, note that as many as five quantities can pos- sibly be involved in any problem about constant acceleration — namely, x − x 0 , v, t, a, and v 0 . Usually, one of these quantities is not involved in the problem, either as a given or as an unknown. We are then presented with three of the remaining quantities and asked to find the fourth. Equations 2-11 and 2-15 each contain four of these quantities, but not the same four. In Eq. 2-11, the “missing ingredient” is the displacement x − x 0 . In Eq. 2-15, it is the velocity v. These two equations can also be combined in three ways to yield three additional equations, each of which involves a different “missing variable.” First, we can eliminate t to obtain v 2 = v 0 2 + 2a(x − x 0 ). (2-16) This equation is useful if we do not know t and are not required to find it. Second, we can eliminate the acceleration a between Eqs. 2-11 and 2-15 to produce an equation in which a does not appear: x − x 0 = 1 2 (v 0 + v)t. (2-17) Finally, we can eliminate v 0 , obtaining x − x 0 = vt − 1 2 at 2 . (2-18) Note the subtle difference between this equation and Eq. 2-15. One involves the initial velocity v 0 ; the other involves the velocity v at time t. Table 2-1 lists the basic Constant Acceleration Equations (Eqs. 2-11 and 2-15) as well as the specialized equations that we have derived. To solve a simple constant acceleration problem, you can usually use an equation from this list (if you have the list with you). Choose an equation for which the only unknown variable is the vari- able requested in the problem. A simpler plan is to remember only Eqs. 2-11 and 2-15, and then solve them as simultaneous equations whenever needed.
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-202 2.3 | One-Dimensional Motion with Constant Acceleration 43 or v 5 v 0 1 at (for constant a) [2.6] Equation 2.6 states that the acceleration a steadily changes the initial velocity steadily changes the initial velocity v 0 by an amount at. For example, if a car starts with a velocity of 12.0 m/s to the right and accelerates to the right with a 5 16.0 m/s 2 , it will have a velocity of 114 m/s after 2.0 s have elapsed: v 5 v 0 1 at 5 1 2.0 m/s 1 (6.0 m/s 2 )(2.0 s) 5 114 m/s The graphical interpretation of v is shown in Figure 2.15b. The velocity varies is shown in Figure 2.15b. The velocity varies linearly with time according to Equation 2.6, as it should for constant acceleration. Because the velocity is increasing or decreasing uniformly with time, we can y with time, we can y express the average velocity in any time interval as the arithmetic average of the initial velocity v 0 and the final velocity v : v 5 v 0 0 1 v 2 (for constant a) [2.7] Remember that this expression is valid only when the acceleration is constant, in which case the velocity increases uniformly. We can now use this result along with the defining equation for average veloc- ity, Equation 2.2, to obtain an expression for the displacement of an object as a function of time. Again, we choose t i 5 0 and t f t f t f 5 t, and for convenience, we write Dx 5 x f x f x f 2 x i 5 x 2 x 0 . This results in Dx 5 v t 5 a v 0 0 1 v 2 b t Dx 5 1 2 1 v 0 0 1 v 2 t (for constant a) [2.8] We can obtain another useful expression for displacement by substituting the equation for v (Eq. 2.6) into Equation 2.8: (Eq. 2.6) into Equation 2.8: Dx 5 1 2 1 v 0 0 1 v 0 0 1 at 2 t D x 5 v 0 0 t 1 1 2 a t 2 (for constant a) [2.9] This equation can also be written in terms of the position x, since Dx 5 x 2 x 0 .
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
(2.4.8) These are not valid when the acceleration is not constant. Free-Fall Acceleration An important example of straight- line motion with constant acceleration is that of an object rising or falling freely near Earth’s surface. The constant- acceleration equations describe this motion, but we make two changes in notation: (1) We refer the motion to the vertical y axis with +y vertically up; (2) we replace a with −g, where g is the magnitude of the free-fall acceleration. Near Earth’s surface, g = 9.8 m/s 2 (= 32 ft/s 2 ). 1 Figure 2.1 gives the velocity of a particle moving on an x axis. What are (a) the initial and (b) the final directions of travel? (c) Does the particle stop momentarily? (d) Is the acceleration positive or negative? (e) Is it constant or varying? 2 Figure 2.2 gives the accel- eration a(t) of a Chihuahua as it chases a German shepherd along an axis. In which of the time periods indicated does the Chihuahua move at constant speed? a A B C D E F G H t FIGURE 2.2 Question 2. 3 Figure 2.3 shows four paths along which objects move from a starting point to a final point, all in the same time interval. The paths pass over a grid of equally spaced straight lines. Rank the paths according to (a) the average velocity of the objects and (b) the average speed of the objects, great- est first. 4 Figure 2.4 is a graph of a particle’s position along an x axis versus time. (a) At time t = 0, what is the sign of the particle’s position? Is the particle’s velocity positive, negative, or 0 at (b) t = 1 s, (c) t = 2 s, and (d) t = 3 s? (e) How many times does the parti- cle go through the point x = 0? 5 Figure 2.5 gives the velocity of a particle moving along an axis. Point 1 is at the highest point on the curve; point 4 is at the low- est point; and points 2 and 6 are at the same height.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
When ∆t becomes infinitesimally small, the average acceleration becomes equal to the 48 CHAPTER 2 Kinematics in One Dimension instantaneous acceleration a → , as indicated in Equation 2.5. Acceleration is the rate at which the velocity is changing. a → = ∆v → ∆t (2.4) a → = lim ∆ t →0 ∆ v → ∆t (2.5) 2.4 Equations of Kinematics for Constant Acceleration/2.5 Applications of the Equations of Kinematics The equations of kinematics apply when an object moves with a constant acceleration along a straight line. These equa- tions relate the displacement x − x 0 , the acceleration a, the final velocity υ, the initial velocity υ 0 , and the elapsed time t − t 0 . Assuming that x 0 = 0 m at t 0 = 0 s, the equations of kinematics are as shown in Equations 2.4 and 2.7–2.9. υ = υ 0 + at (2.4) x = 1 2 (υ 0 + υ) t (2.7) x = υ 0 t + 1 2 at 2 (2.8) υ 2 = υ 2 0 + 2ax (2.9) 2.6 Freely Falling Bodies In free-fall motion, an object experiences neg- ligible air resistance and a constant acceleration due to gravity. All objects at the same location above the earth have the same acceleration due to gravity. The acceleration due to gravity is directed toward the center of the earth and has a magnitude of approximately 9.80 m/s 2 near the earth’s surface. 2.7 Graphical Analysis of Velocity and Acceleration The slope of a plot of position versus time for a moving object gives the object’s velocity. The slope of a plot of velocity versus time gives the object’s acceleration. Note to Instructors: The numbering of the questions shown here reflects the fact that they are only a representative subset of the total number that are available online. However, all of the questions are available for assignment via WileyPLUS. Section 2.1 Displacement 1. What is the difference between distance and displacement? (a) Distance is a vector, while displacement is not a vector. (b) Displacement is a vector, while distance is not a vector.
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 44 TOPIC 2 | Motion in One Dimension Unless otherwise noted, all content on this page is © Cengage Learning. PROBLEM-SOLVING STRATEGY Motion in One Dimension at Constant Acceleration The following procedure is recommended for solving problems involving accelerated motion. 1. Read the problem. 2. Draw a diagram, choosing a coordinate system, labeling initial and final points, and indicating directions of velocities and accelerations with arrows. 3. Label all quantities, circling the unknowns. Convert units as needed. 4. Equations from Table 2.4 should be selected next. All kinematics problems in this topic can be solved with the first two equations, and the third is often convenient. 5. Solve for the unknowns. Doing so often involves solving two equations for two unknowns. 6. Check your answer, using common sense and estimates. Most of these problems reduce to writing the kinematic equations from Table 2.4 and then substituting the correct values into the constants a, v 0 , and x 0 from the given information. Doing this produces two equations—one linear and one quadratic— for two unknown quantities. Tip 2.8 Pigs Don’t Fly After solving a problem, you should think about your answer and decide whether it seems rea- sonable. If it isn’t, look for your mistake! Table 2.4 Equations for Motion in a Straight Line Under Constant Acceleration Equation Information Given by Equation v 5 v 0 1 at Velocity as a function of time Dx 5 v 0 t 1 1 2 at 2 Displacement as a function of time v 2 5 v 0 2 1 2a Dx Velocity as a function of displacement Note: Motion is along the x - axis. At t 5 0, the velocity of the particle is v 0 . EXAMPLE 2.4 THE BATHURST 1000 GOAL Apply the basic kinematic equations. PROBLEM A race car starting from rest accelerates at a constant rate of 5.00 m/s 2 .
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Average velocity is a vector that has the same direction as the displacement. When the elapsed time becomes infinitesimally small, the average velocity becomes equal to the instantaneous velocity v B , the velocity at an instant of time, as indicated in Equation 2.3. 2.3 Acceleration The average acceleration a B is a vector. It equals the change D v B in the velocity divided by the elapsed time Dt, the change in the velocity being the final minus the initial velocity; see Equation 2.4. When Dt becomes infinitesimally small, the average acceleration becomes equal to the instantaneous acceleration a B , as indicated in Equation 2.5. Acceleration is the rate at which the velocity is changing. 2.4 Equations of Kinematics for Constant Acceleration/2.5 Applications of the Equations of Kine- matics The equations of kinematics apply when an object moves with a constant acceleration along a straight line. These equations relate the displacement x 2 x 0 , the acceleration a, the final velocity v, the initial velocity v 0 , and the elapsed time t 2 t 0 . Assuming that x 0 5 0 m at t 0 5 0 s, the equations of kinematics are as shown in Equations 2.4 and 2.7–2.9. 2.6 Freely Falling Bodies In free-fall motion, an object experiences negligible air resistance and a constant acceleration due to gravity. All objects at the same location above the earth have the same acceleration due to gravity. The acceleration due to gravity is directed toward the center of the earth and has a magnitude of approximately 9.80 m/s 2 near the earth’s surface. 2.7 Graphical Analysis of Velocity and Acceleration The slope of a plot of position versus time for a moving object gives the object’s velocity. The slope of a plot of velocity versus time gives the object’s acceleration.
No longer available |Learn morePhysics for Scientists and Engineers
Foundations and Connections, Extended Version with Modern Physics
- Debora Katz(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Motion diagram 2. Sketch with a coordinate system GENERAL PROBLEM-SOLVING STRATEGIES 3. Position-, velocity-, or acceleration-versus-time graphs Not all problems involve constant acceleration. When the acceleration changes, you must use calculus (not Table 2.4) to solve the problem. ★ Major Concepts—cont'd 8. Average acceleration: change in a particle’s velocity over some time interval. a u av, x ; Dv u x Dt 5 Dv x Dt d ˆ 5 v xf 2 v xi t f 2 t i d ˆ a u av, y ; Dv y Dt e ˆ a u av, z ; Dv z Dt k ˆ (2.6) Instantaneous acceleration (also known as simply acceleration): the average acceleration as Dt ap- proaches zero. a u x ; lim Dt S0 Dv u x Dt 5 d v u x dt 5 dv x dt d ˆ a u y ; dv y dt e ˆ a u z ; dv z dt k ˆ (2.7) ▲ Special Cases 1. In the special case of constant acceleration the ac- celeration has the same magnitude and direction at all times, greatly simplifying the mathematical analysis. 2. An object accelerating solely under the influence of gravity is said to be in free fall. An object in free fall near the surface of the Earth has acceleration of magnitude g 5 9.81 m / s 2 . As part of the INTERPRET and ANTICIPATE procedure: Draw a sketch that includes a coordinate system. There are four parts to the SOLVE procedure: 1. Start by listing the six kinematic variables (initial position, final position, initial velocity, final velocity, acceleration, and time). Often, the initial and final positions are combined as the displacement, so your list typically includes five variables. Include the sign of each value. Write “need” next to any variable you need to solve for and write “not needed” next to any PROBLEM-SOLVING STRATEGY variable that you don’t know and don’t need to find. If the problem involves free fall, remember that the acceleration has a magnitude of g 5 9.81 m/s 2 and points downward toward the center of the Earth. 2. Once you have listed the parameters, you may use the Constant Acceleration Equations (Table 2.4).
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
