Physics
Uniformly Accelerated Motion
Uniformly Accelerated Motion refers to the motion of an object with a constant acceleration. In this type of motion, the velocity of the object changes by the same amount in equal time intervals. This results in a linear increase or decrease in velocity over time, leading to a parabolic path for the object's motion.
Written by Perlego with AI-assistance
Related key terms
1 of 5
10 Key excerpts on "Uniformly Accelerated Motion"
eBook - PDFApplied Mathematics
Made Simple
- Patrick Murphy(Author)
- 2014(Publication Date)
- Butterworth-Heinemann(Publisher)
If the increase or decrease in the velocity is the same for every second of the motion, then the acceleration is said to be uniform. It should be noted that acceleration has magnitude, direction, and sense. When the velocity of a body is increasing, the motion of the body is said to be 'accelerating'; when the velocity is decreasing, the motion is said to be 'retarding'. Again we may associate a positive or negative sign with each of these situations, in fact we refer to a retardation as being a negative acceleration. If a velocity is measured in metres per second, then acceleration (i.e. the change in velocity per second) may be measured in metres per second per second. Thus, an acceleration of 3 metres per second per second means that the magnitude of the velocity of the body is being increased by 3 metres per second every second. Since the motion takes place in a straight line, the acceleration, velocity, and displacement of a body are completely specified by three positive or negative numbers. We shall therefore speak of velocity-time and displacement-time graphs instead of speed-time and distance-time graphs. Motion with Uniform Acceleration in a Straight Line 5 1 Example: A body starting from rest moves in a straight line with a uniform accelera-tion of 5 metres per second per second for 6 seconds. Draw a velocity-time graph for the motion; from it find (a) the velocity at / = 6 seconds, and (b) the distance travelled in 6 seconds.
eBook - PDFWorkshop Physics Activity Guide Module 1
Mechanics I
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Name Section Date UNIT 4: MOTION WITH CONSTANT ACCELERATION chattereye / Adobe Stock Physicists often describe objects or situations using models. A model can be a physical, scaled- down version of an object, a visual representation of such an object on a computer screen, or even a set of mathematical equations. For example, it is convenient to model the actual position and velocity of an object as a function of time using equations. Equations can be readily graphed, rearranged, and studied in ways that help illuminate the underlying motion of an object. In this unit, we will learn how to create mathematical models of objects moving with constant acceleration. 100 WORKSHOP PHYSICS ACTIVITY GUIDE UNIT 4: MOTION WITH CONSTANT ACCELERATION OBJECTIVES 1. To recognize position-time, velocity-time, and acceleration-time graphs for constantly accelerated motion. 2. To use mathematical models to describe one-dimensional motion with constant acceleration. 3. To learn to use the kinematic equations for objects moving with constant acceleration. 4.1 OVERVIEW In this unit, we will continue to study the motion of a low-friction cart propelled by a fan. In the first set of activities we will observe acceleration as the cart speeds up, slows down, and turns around. We then derive a useful set of formulas known as the kinematic equations, which describe how the position and velocity of an object change when undergoing a constant acceleration. These equations can be used to model the motion of our accelerating cart. A wide range of everyday motions involve constant (or near constant) acceleration, and we finish this unit by using the kinematic equations to describe the motion of various objects. UNIT 4: MOTION WITH CONSTANT ACCELERATION 101 SPEEDING UP, SLOWING DOWN, AND TURNING AROUND 4.2 DESCRIBING VELOCITIES AND ACCELERATIONS Describing Velocities The term speed is familiar to most students as a description of how fast or slow an object is moving.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
(Answer) D θ 0 v 0 y x Launch Water pool (a) θ 0 v 0 v 0y v 0x θ 0 v v y v 0x (b) (c) Landing velocity Launch velocity Figure 4-15 (a) Launch from a water slide, to land in a water pool. The velocity at (b) launch and (c) landing. Additional examples, video, and practice available at WileyPLUS 67 Uniform Circular Motion A particle is in uniform circular motion if it travels around a circle or a circular arc at constant (uniform) speed. Although the speed does not vary, the particle is accelerating because the velocity changes in direction. Figure 4-16 shows the relationship between the velocity and acceleration vectors at various stages during uniform circular motion. Both vectors have constant magnitude, but their directions change continuously. The velocity is always directed tangent to the circle in the direction of motion. The acceleration is always directed radially inward. Because of this, the acceleration associated with uniform circular motion is called a centripetal (meaning “center seeking”) acceleration. As we prove next, the magnitude of this acceleration a → is a = v 2 r (centripetal acceleration), (4-34) where r is the radius of the circle and v is the speed of the particle. In addition, during this acceleration at constant speed, the particle travels the circumference of the circle (a distance of 2πr) in time T = 2πr v (period). (4-35) T is called the period of revolution, or simply the period, of the motion. It is, in general, the time for a particle to go around a closed path exactly once. Proof of Eq. 4-34 To find the magnitude and direction of the acceleration for uniform circular motion, we consider Fig. 4-17. In Fig. 4-17a, particle p moves at constant speed v around a circle of radius r. At the instant shown, p has coordinates x p and y p . Recall from Module 4-2 that the velocity v → of a moving particle is always tangent to the particle’s path at the particle’s position.
eBook - PDFGeneral Engineering Science in SI Units
The Commonwealth and International Library: Mechanical Engineering Division
- G. W. Marr, N. Hiller(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
The appropriate velocity-time graph is shown in Fig. 2.8. The graph in Fig. 2.9 also represents uniformly accelerated 41 GENERAL ENGINEERING SCIENCE IN SI UNITS t, u Time FIG. 2.9. motion starting from rest. In this case, the constant acceleration, a = — = gradient of graph. h — h If the body should start with an initial velocity ν λ and move with constant acceleration, a 9 the graph is again a straight line, but instead of passing through the origin, it cuts the vertical axis in Vi (Fig. 2.10). As before, the gradient of the line is equal to the constant acceleration, a. After t seconds the velocity is ^2 = Vi+at. V 2 0 T ime FIG. 2.10. 42 VELOCITY AND ACCELERATION 2.6. Average Velocity during Uniformly Accelerated Motion For any period during which the acceleration is constant, the average velocity is the average of the initial and final velocities, i.e. v = {vi+v 2 ) where v is the average velocity (Fig. 2.11). The distance travelled in time / when the average velocity is v is s = v.t Λ S = γ(Όι + Ό 2 )ί. But, as previously shown, for constant acceleration v 2 = vi + at. Λ s= iOi-Kvi + aOl' = K2t>i + a/)/ = vit + àt 2 . In the particular case where the body starts from rest, v } - 0: s = ia/ 2 . 43 GENERAL ENGINEERING SCIENCE IN SI UNITS 2.7. Other Relationships for Uniformly Accelerated Motioh We have already established the formula / or v 2 — Vi = at (1) and the formula s = (vi + v 2 )t, from which we obtain vi + v 2 = — . (2) Combining (1) and (2) we can write Is {v 2 — V){v2 + v{) = atX — = las i.e. v — v = las This formula relates distance and the initial and final velocities during a period of constant acceleration without direct reference to the length of the time interval. A special case is that of a body starting from rest. Then Vi = 0 and if the final velocity be i we obtain the simple formula v 2 = las. EXAMPLE. A body, starting from rest, moves with a constant acceleration of 0-25 m/s 2 .
eBook - PDFGeneral Engineering Science in SI Units
In Two Volumes
- G. W. Marr, N. Hiller(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
Time, s FIG. 2.8. For example, if the body moves with a constant acceleration of 5 m/s 2 , this means that during each successive second the velocity will increase by 5 m/s. The appropriate velocity-time graph is shown in Fig. 2.8. The graph in Fig. 2.9 also represents uniformly accelerated 41 FIG. 2.7. GENERAL ENGINEERING SCIENCE IN SI UNITS motion starting from rest. In this case, the constant acceleration, a = = gradient of graph. t2-h If the body should start with an initial velocity v ± and move with constant acceleration, a, the graph is again a straight line, but instead of passing through the origin, it cuts the vertical axis in v x (Fig. 2.10). As before, the gradient of the line is equal to the constant acceleration, a. After t seconds the velocity is ^2 = νχ+at. 42 FIG. 2.9. FIG. 2.10. VELOCITY AND ACCELERATION 2.6. Average Velocity during Uniformly Accelerated Motion For any period during which the acceleration is constant, the average velocity is the average of the initial and final velocities, i.e. v = (v x +v 2 ) where v is the average velocity (Fig. 2.11). The distance travelled in time t when the average velocity is v is s = v.t ;. s = (v x + v2)t. But, as previously shown, for constant acceleration v 2 = Vi + at. .'. ■* = Uvi + (vi +at)]t = i2v x + at)t = Vit + iat*. In the particular case where the body starts from rest, Vj =■ 0: s = at*. 43 FIG. 2.11. GENERAL ENGINEERING SCIENCE IN SI UNITS 2.7. Other Relationships for Uniformly Accelerated Motion We have already established the formula a = t or t>2 —0i = at (1) and the formula s = γ(νι + v 2 )t, from which we obtain #1 + ^2 = — . (2) Combining (1) and (2) we can write Is (v 2 — v 1 )(v2 + v{) = atX — = las i.e. v — v = las This formula relates distance and the initial and final velocities during a period of constant acceleration without direct reference to the length of the time interval. A special case is that of a body starting from rest.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
• The time of flight of a projectile launched with initial vertical velocity v 0y on an even surface is given by T tof = 2(v 0 sin θ) g . This equation is valid only when the projectile lands at the same elevation from which it was launched. • The maximum horizontal distance traveled by a projectile is called the range. Again, the equation for range is valid only when the projectile lands at the same elevation from which it was launched. 4.4 Uniform Circular Motion • Uniform circular motion is motion in a circle at constant speed. • Centripetal acceleration a → C is the acceleration a particle must have to follow a circular path. Centripetal acceleration always points toward the center of rotation and has magnitude a C = v 2 /r. • Nonuniform circular motion occurs when there is tangential acceleration of an object executing circular motion such that the speed of the object is changing. This acceleration is called tangential acceleration a → T . The magnitude of tangential acceleration is the time rate of change of the magnitude of the velocity. The tangential acceleration vector is tangential to the circle, whereas the centripetal acceleration vector points radially inward toward the center of the circle. The total acceleration is the vector sum of tangential and centripetal accelerations. • An object executing uniform circular motion can be described with equations of motion. The position vector of the object is r → (t) = A cos ωt i ^ + A sin ωt j ^ , where A is the magnitude | r → (t) | , which is also the radius of the circle, and ω is the angular frequency. 4.5 Relative Motion in One and Two Dimensions • When analyzing motion of an object, the reference frame in terms of position, velocity, and acceleration needs to be specified. • Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies with the choice of reference frame. Chapter 4 | Motion in Two and Three Dimensions 197
eBook - ePub- Hiqmet Kamberaj(Author)
- 2021(Publication Date)
- De Gruyter(Publisher)
h, then it freely falls down until hits the ground with a speed (since the air resistance is ignored):(4.41)v =2 h g=v i.For, we obtainθ i=45 ∘(4.42)This corresponds to the maximum displacement along the horizontal direction.x=v i 2gh==v i 2( 2 / 2 )22 g.v i 24 gFor, we do not obtain a projectile motion, but a straight-line motion with constant velocityθ i=0 ∘v i, if we ignore friction with horizontal surface.4.4 Uniform circular motion
Another example of planar motion is circular motion, which represents the motion of a particle along a circular path.Uniform circular motion
The motion of a particle along a circular path is defined by the velocity and the acceleration of that particle. The velocity vector of the particle is tangent to the circular trajectory at every instant. The velocity is a vector quantity, and hence, it has a magnitude and direction. By definition, the circular motion is uniform if the velocity vector’s magnitude is constant, but its direction changes as the particle moves along the circular path.Centripetal acceleration
Since the direction of the velocity changes, this causes the particle to accelerate along the circle. By definition, that acceleration is called centripetal acceleration, which stands for “acceleration seeking a center”. Therefore, the acceleration vector always has a direction towards the center of the circle. Fig. 4.8 presents the velocity and acceleration vectors as the particle moves along a circular path from point P to the point Q with an angleΔ θin a circle with radius r
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
(4.3.2) In unit-vector notation, a → = a x ˆ i + a y ˆ j + a z k ̂ , (4.3.3) where a x = dv x /dt, a y = dv y /dt, and a z = dv z /dt. Review & Summary Projectile Motion Projectile motion is the motion of a par- ticle that is launched with an initial velocity v → 0 . During its flight, the particle’s horizontal acceleration is zero and its vertical acceleration is the free-fall acceleration ‒g. (Upward is taken to be a positive direction.) If v → 0 is expressed as a magnitude (the speed v 0 ) and an angle θ 0 (measured from the horizontal), the particle’s equations of motion along the horizontal x axis and vertical y axis are x − x 0 = ( v 0 cos θ 0 )t, (4.4.3) y − y 0 = ( v 0 sin θ 0 )t − 1 _ 2 gt 2 , (4.4.4) v y = v 0 sin θ 0 − gt, (4.4.5) v y 2 = ( v 0 sin θ 0 ) 2 − 2g(y − y 0 ). (4.4.6) The trajectory (path) of a particle in projectile motion is parabolic and is given by y = (tan θ 0 )x − gx 2 ____________ 2( v 0 cos θ 0 ) 2 , (4.4.7) if x 0 and y 0 of Eqs. 4.4.3 to 4.4.6 are zero. The particle’s horizontal range R, which is the horizontal distance from the launch point to the point at which the particle returns to the launch height, is R = v 0 2 __ g sin 2θ 0 . (4.4.8) Uniform Circular Motion If a particle travels along a circle or circular arc of radius r at constant speed v, it is said to be in uniform circular motion and has an acceleration a → of constant magnitude a = v 2 ___ r . (4.5.1) The direction of a → is toward the center of the circle or circular arc, and a → is said to be centripetal. The time for the particle to complete a circle is T = 2πr ____ v . (4.5.2) T is called the period of revolution, or simply the period, of the motion. Relative Motion When two frames of reference A and B are moving relative to each other at constant velocity, the velocity of a particle P as measured by an observer in frame A usually differs from that measured from frame B.
eBook - PDF- James Shipman, Jerry Wilson, Charles Higgins, Bo Lou, James Shipman(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2.3 Acceleration 35 When any of these changes occur, an object is accelerating. Examples are 1. a car speeding up (or slowing down) while traveling in a straight line, 2. a car rounding a curve at a constant speed, and 3. a car speeding up (or slowing down) while rounding a curve. Acceleration is defined as the time rate of change of velocity. Taking the symbol Δ (delta) to mean “change in,” the equation for average acceleration (a) can be written as average acceleration 5 change in velocity time for change to occur a 5 Dv Dt 5 v f 2 v o t 2.2 The change in velocity (Δv) is the final velocity v f minus the original velocity v o . Also, the interval Δt is commonly written as t (Δt 5 t 2 t o 5 t), with t o taken to be zero (t is understood to be an interval). v o is not zero if the car is initially moving. The units of acceleration in the SI are meters per second per second, (m/s)/s, or meters per second squared, m/s 2 . These units may be confusing at first. Keep in mind that an acceleration is a measure of a change in velocity during a given time period. Consider a constant acceleration of 9.8 m/s 2 . This value means that the velocity changes by 9.8 m/s each second. Thus, for straight-line motion, as the number of seconds increases, the velocity goes from 0 to 9.8 m/s during the first second, to 19.6 m/s (that is, 9.8 m/s 1 9.8 m/s) during the second second, to 29.4 m/s (that is, 19.6 m/s 1 9.8 m/s) during the third second, and so forth, adding 9.8 m/s each second. This sequence is illustrated in ● Fig. 2.7 for an object that falls with a constant acceleration due to gravity of 9.8 m/s 2 . Because the velocity increases, the distance traveled by the falling object each second also increases, but not uniformly.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
The direction of a → is toward the center of the circle or circular arc, and a → is said to be centripetal. The time for the particle to complete a circle is T = 2πr ___ v . T is called the period of revolution, or simply the period, of the motion. Figure 4.5.1 Velocity and acceleration vectors for uniform circular motion. v v v a a a The acceleration vector always points toward the center. The velocity vector is always tangent to the path. Uniform Circular Motion A particle is in uniform circular motion if it travels around a circle or a circular arc at constant (uniform) speed. Although the speed does not vary, the particle is accelerating because the velocity changes in direction. Figure 4.5.1 shows the relationship between the velocity and acceleration vectors at various stages during uniform circular motion. Both vectors have constant magnitude, but their directions change continuously. The velocity is always directed tangent to the circle in the direction of motion. The acceleration is always directed radially inward. Because of this, the acceleration associated with uniform circular motion is called a centripetal (meaning “center seeking”) acceleration. As we prove next, the magnitude of this acceleration a → is a = v 2 __ r (centripetal acceleration), (4.5.1) where r is the radius of the circle and v is the speed of the particle. In addition, during this acceleration at constant speed, the particle travels the circumference of the circle (a distance of 2πr) in time T = 2πr ___ v (period). (4.5.2) T is called the period of revolution, or simply the period, of the motion. It is, in general, the time for a particle to go around a closed path exactly once. Proof of Eq. 4.5.1 To find the magnitude and direction of the acceleration for uniform circular motion, we consider Fig. 4.5.2. In Fig. 4.5.2a, particle p moves at constant speed v around a circle of radius r. At the instant shown, p has coordinates x p and y p .
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.
