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.
6 Key excerpts on "Constant Acceleration Equations"
eBook - ePub- Alan Hendrickson(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...2 The Equations of Constant Acceleration DOI: 10.4324/9780080557540-2 Constant Acceleration Acceleration is a measure of how rapidly an object speeds up or slows down, or more specifically, how much its velocity changes over some given interval of time. Acceleration can be measured both at a point in time, the result being called instantaneous acceleration, or over some finite time interval, called an average acceleration. Constant acceleration exists whenever both measures of acceleration remain the same for some period of time. During these time periods, average acceleration, ā, and instantaneous acceleration, a, are equal, and both hold at one value regardless of the length of the time interval chosen for Δ t : a = a ¯ = Δ v Δ t i f a c c e l e r a t i o n i s c o n s t a n t In a typical scenery move, there are three key periods where acceleration can be considered constant: during acceleration from zero on up to top speed, during travel at a constant top speed, and during deceleration to a stop (see motion profiles in Figure 2.1). During the constant velocity portion, acceleration, a, will be zero because velocity does not change. (True too before and after the move, but since nothing is moving then, it is of no interest to us here.) During a constant acceleration or deceleration, the change of velocity versus time is a fixed number regardless of the value chosen for Δ t. For three arbitrary values of t 1 and t 2 in the example in Figure 2.1, acceleration calculates out to the same value: Figure 2.1 Typical constant acceleration motion. profiles f o r t 1 = 1 a n d t 2 = 2 a ¯ = v 2 − v 1 t 2 − t 1 = 1 − 0 2 − 1 = 1 f t / s e c 2 f o r t 1 = 1 a n d t 2 = 5 a ¯ = v 2 − v 1 t 2 − t 1 = 4 − 0 5 − 1 = 1 f t /[--=PL...
eBook - ePub- W. Bolton(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...If the velocity changes from 5 m/s to 2 m/s in 10 s then the average acceleration over that time is (22 5)/10 5 20.3 m/s 2, i.e. it is a retardation. 11 A constant or uniform acceleration occurs when the velocity changes by equal amounts in equal intervals of time, however small the time interval. Thus an object with a constant acceleration of 5 m/s 2 in a particular direction for a time of 30 s will change its velocity by 5 m/s in the specified direction in each second of its motion. 4.2 Straight line motion The equations that are derived in the following discussion all relate to uniformly accelerated motion in a straight line. If u is the initial velocity, i.e. at time t = 0, and v the velocity after some time t, then the change in velocity in the time interval t is (v - u). Hence the acceleration a is (v 2 u)/ t. Rearranging this gives: v = u + at [Equation 1] If the object, in its straight- line motion, covers a distance s in a time t, then the average velocity in that time interval is s/t. With an initial velocity of u and a final velocity of v at the end of the time interval, the average velocity is (u + v)/2. Hence s/t 5 (u 1 v)/2 and so: Substituting for v by using the equation v = u + at gives: Consider the equation v = u + at. Squaring both sides of this equation gives: Hence, substituting for the bracketed term using equation 2: v 2 = u 2 + 2as [Equation 3] The equations [1], [2] and [3] are referred to as the equations for straight- line motion. The following examples illustrate their use in solving engineering problems. Example An object moves in a straight line with a uniform acceleration. If it starts from rest and takes 12 s to cover 100 m, what is the acceleration? If it continues with the same acceleration, how long will it take to cover the next 100 m and what will be its velocity after the 200 m? For the first 100 m, we have u = 0, s = 100 m, t = 12 s and are required to obtain a...
eBook - ePub- A. D. Johnson(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...The motion can be expressed graphically as shown in Figure 2.5. Fig. 2.5 Uniform acceleration shown on a velocity–time graph. Referring to Figure 2.5, the following observations can be made: change of velocity = v 2 − v 1 This change of velocity can be related to the acceleration because acceleration = a = rate of change of velocity = changeofvelocity timetaken = v 2 − v 1 t and transposing gives v 2 = v 1 + a t (2.2) This equation can be compared directly with the diagram in Figure 2.5. Figure 2.5 shows that the velocity increases at a uniform rate between v 1 and v 2. The average velocity can therefore be described as v ¯ = (v 1 + v 2) 2 and if s is the distance travelled during that period, equation (2.1) may be used in the form s = v t = average velocity × time In other words s = v ¯ × t and s = (v 1 + v 2) t 2 (2.3) From equation (2.2), it is possible to substitute for υ 2 in equation (2.3) : s = (v 1 + v 1 + a t) t 2 and rearranging. gives s = v 1 t + 1 2 a t 2 (2.4) The above equations are adequate for most purposes but there often arise situations where time t is the unknown. Since equations (2.2), 2.3 and (2.4) all possess t an expression is now needed for v 2 in terms of v 1, a and s, which excludes t. This is done by squaring both sides of equation (2.2) and substituting for s from equation (2.4). From equation (2.2) v 2 = v 1 + a t and squaring gives v 2 2 = (v 1 + a t) 2 so that v 2 2 = v 1 2 + 2 v 1 a t + a 2 t 2 or v 2 2 = v 1 2 + 2 a (v 1 t + 1 2 a t 2) But (v 1 t + 1 2 a t 2) is equal to s from equation. (2.4) : v 2 2 = v 1 2 + 2 a s (2.5) The equations (2.2), 2.3, 2.4 and (2.5), derived above, form the basis of all motion studies at this level and can easily be manipulated to cope with a range of situations, including falling bodies and rotary motion, as explained later. Example 2.3 A train has a uniform acceleration of 0.2 m/s 2 along a straight track...
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Chapter 18 Acceleration Why it is important to understand: Acceleration Acceleration may be defined as a ‘change in velocity’. This change can be in the magnitude (speed) of the velocity or the direction of the velocity. In daily life we use acceleration as a term for the speeding up of objects and decelerating for the slowing down of objects. If there is a change in the velocity, whether it is slowing down or speeding up, or changing its direction, we say that the object is accelerating. If an object is moving at constant speed in a circular motion – such as a satellite orbiting the earth – it is said to be accelerating because change in direction of motion means its velocity is changing even if speed may be constant. This is called centripetal (directed towards the centre) acceleration. On the other hand, if the direction of motion of the object is not changing but its speed is, this is called tangential acceleration. If the direction of acceleration is in the same direction as that of velocity then the object is said to be speeding up or accelerating. If the acceleration and velocity are in opposite directions then the object is said to be slowing down or decelerating. An example of constant acceleration is the effect of the gravity of earth on an object in free fall. Measurement of the acceleration of a vehicle enables an evaluation of the overall vehicle performance and response. Detection of rapid negative acceleration of a vehicle is used to detect vehicle collision and deploy airbags. The measurement of acceleration is also used to measure seismic activity, inclination and machine vibration...
eBook - ePub- Ronald J. Anderson(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...Rockets, for instance, have a fairly large rate of change of mass as fuel is consumed during takeoff and must be considered. The majority of mechanical systems on earth are composed of rigid or flexible bodies that do not suffer a mass change during their motions. For this reason, Equation 2.2 is most often used in the form, (2.3) where is defined to be the acceleration of the particle. (2.4) Both the velocity and the acceleration of the particle relate its motion to an inertial reference frame and they are termed the absolute velocity and absolute acceleration respectively. 2.4 Deriving Equations of Motion for Particles Newton's laws provide a very convenient method for deriving the equations of motion of simple systems. Equations of motion are the differential equations that, when solved, can be used to predict the response of the system to a set of applied forces. The procedure for deriving the equations has only four steps and, if they are followed, the derivations are very straightforward. The steps are, Kinematics – choose the coordinates (also known as degrees of freedom) to be used to describe the motion and derive expressions for the absolute velocities and accelerations of the masses under consideration. Use the methods of Chapter 1. Free body diagrams (FBDs) – sketch the masses under consideration as if they are in space with no forces acting on them. Then add to the sketches all of the externally applied forces acting on the masses. Also add the internal forces of interaction between the masses, being sure that they act in equal and opposite pairs as stipulated by Newton's third law. The FBDs should show the positive sense of the accelerations derived in step 1. Force balance equations (Newton's second law) – using the FBDs, equate the vector sum of forces on each body to its mass multiplied by its vector acceleration. Manipulate and solve the equations – the first three steps will lead to a set of equations with a set of unknowns...
eBook - ePubAutomotive Accident Reconstruction
Practices and Principles, Second Edition
- Donald E. Struble, John D. Struble(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
...Therefore what-if questions may focus on the effects of varying the perception–reaction time from say a ½ to 2½ sec. Constant Acceleration The simplest kind of motion is at constant velocity, for which “rest” is simply a special case. Most often, though, time–distance studies involve vehicles that are being accelerated. The type of acceleration that is simplest to analyze is a constant, for which constant velocity (zero acceleration) is a special case. If a trajectory can be divided into segments with constant acceleration (as has been done in Chapters 3 and 4), the resulting analysis will cover the vast majority of questions asked of the reconstructionist. Generally, it is not necessary to make the segments small, but of course the velocity and the travel distance need to be continuous functions over an entire path. It is possible to write a general-purpose computer program in which different formulas are used in various segments, depending on what is known and what is unknown in any given segment. One can even do a forward-directed or a backward-directed calculation, depending on whether the velocity at the end is known (a final-value problem) or the velocity at the beginning is known (an initial-value problem). Obviously, a lot of IF-THEN-ELSE constructs must be employed. In all cases, the object is to wind up knowing the velocity at each of the segment boundaries, otherwise known as the key points. In addition, the user should have access to both the elapsed time (which is a forward measure) and the remaining time (which is a backward measure). Often the final event is a collision, though it could also be a vehicle coming to rest or even something else. Similarly, the user should know or be able to calculate the elapsed distance and the remaining distance. In each segment, the user should know or be able to calculate Δ S, the segment length, and Δ t, the segment time. The time–distance study can (and usually does) involve more than one vehicle...
