Acceleration and Time
Acceleration is the rate of change of velocity over time. It is a vector quantity, meaning it has both magnitude and direction. Time is a fundamental concept in mathematics that measures the duration of events and the intervals between them. In the context of acceleration, time is crucial for understanding how velocity changes over a specific period.
5 Key excerpts on "Acceleration and Time"
eBook - ePub- A. D. Johnson(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...2.2 Displacement–time graph. v = s t so that t = s v t = 120000 208.33 = 576s or 9 min 36s 2.2.5 Acceleration When a car sets off from traffic lights the driver depresses the ‘accelerator’ and the car steadily increases velocity. In slowing the car the driver depresses the brake pedal, which can be considered as a ‘decelerator’ This example serves to define the terms acceleration and deceleration; however, other descriptive terms for deceleration are retardation and negative acceleration. Acceleration is the change in velocity compared to advancing time, or a = s t × 1 t = s t 2 where a = acceleration in units of m/s 2. Example 2.2 The velocity of a car on a straight level road increases by 2.0 m/s every second as it accelerates from standstill until it reaches 40 m/s. Tabulate the velocity second by second. Solution The answer can be tabulated as follows: It can be seen that the acceleration is uniform (constant) and as every second ticks by the velocity increases by 2 m/s. The acceleration can, therefore, be seen to be 2 m/s 2. 2.2.6 Velocity–time graphs Acceleration can be defined as the rate of increase of velocity. It can be represented on a velocity–time graph as shown in Figure 2.3. Fig. 2.3 Velocity–time graph. The velocity–time graph is a progression from the displacement–time graph shown in Figure 2.2. Instead of displacement, velocity is measured on the vertical axis. When velocity is plotted against time, the graph line represents acceleration. Figure 2.3 shows a straight graph line indicating uniform acceleration; however, in practice, acceleration can also vary. It should be noted that the area under the graph line represents displacement and can be calculated by determining areas directly from the graph but also by considering equation (2.1) : v = s t or, transposing, s = v × t...
eBook - ePub- Ronald J. Anderson(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...Part I Modeling: Deriving Equations of Motion 1 Kinematics Kinematics is defined as the study of motion without reference to the forces that cause the motion. A proper kinematic analysis is an essential first step in any dynamics problem. This is where the analyst defines the degrees of freedom and develops expressions for the absolute velocities and accelerations of the bodies in the system that satisfy all of the physical constraints. The ability to differentiate vectors with respect to time is a critical skill in kinematic analysis. 1.1 Derivatives of Vectors Vectors have two distinct properties – magnitude and direction. Either or both of these properties may change with time and the time derivative of a vector must account for both. The rate of change of a vector with respect to time is therefore formed from, The rate of change of magnitude. The rate of change of direction. Figure 1.1 A vector changing with time. Figure 1.1 shows the vector that changes after a time increment,, to. The difference between and can be defined as the vector shown in Figure 1.1 and, by the rules of vector addition, (1.1) or, (1.2) Then, using the definition of the time derivative, (1.3) Imagine now that Figure 1.1 is compressed to show only an infinitesimally small time interval,. The components of for the interval are shown in Figure 1.1. They are, A component aligned with the vector. This is a component that is strictly due to the rate of change of magnitude of. The magnitude of is where is the rate of change of length (or magnitude) of the vector. The direction of is the same as the direction of. Let be designated 1 as. A component that is perpendicular to the vector. That is, a component due to the rate of change of direction of the vector. Terms of this type arise only when there is an angular velocity...
eBook - ePubInstant Notes in Sport and Exercise Biomechanics
Second Edition
- Paul Grimshaw, Michael Cole, Adrian Burden, Neil Fowler(Authors)
- 2019(Publication Date)
- Garland Science(Publisher)
...Obviously, such knowledge of individual and comparative performances would have important training implications for both the athlete and the coach. Acceleration is defined as the change in velocity per unit of time and it is usually measured in metres per second squared (m/s 2). This means the velocity of an object will increase/decrease by an amount for every second of its motion. For example, a constant (uniform) acceleration of 2.5 m/s 2 indicates that the body will increase its velocity by 2.5 m/s for every second of its motion (2.5 m/s for 1 second, 5.0 m/s for 2 seconds, 7.5 m/s for 3 seconds and so on). Table A1.3 shows the Microsoft Excel calculations that now include the acceleration data for the 100 m sprint performance used in the previous example. Table A1.2 Microsoft Excel calculations (to two decimal places) of the velocity for each 10 m interval in the 100 m sprint data (d = displacement; dt = time for each 10 m; sum t = cumulative time; v = velocity; sd = standard deviation; max = maximum value; min = minimum value) Table A1.3 Microsoft Excel calculations (to two decimal places) of the acceleration for each 10 m interval in the 100 m sprint (d = displacement; dt = time for each 10 m; sum t = cumulative time; v = velocity; a = acceleration; sd = standard deviation; max = maximum value; min = minimum value) Acceleration can be represented by the...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...It does not take into account direction of motion. Speed at any moment t = c is | v(c) | Acceleration If x (t) is the position of a particle moving along the x -axis, its acceleration The acceleration of the particle measured at any moment t = c, is a (c) = v ′(c) = x ″(c). EXAMPLE 5.19 The position of a particle moving along the x -axis is defined by x (t) = t · sin(2 t). Find the displacement of the particle on the time interval, t = 0 to seconds. SOLUTION The particle has oscillated during the interval, but has returned to its starting position when seconds, so there is no displacement. EXAMPLE 5.20 The position of a particle moving along the x -axis is defined by x (t) = l n (t 2 + 1) for t ≥ 0. If distance is measured in feet and time is in seconds, find the average acceleration on the time interval [1, 3] seconds. SOLUTION If average velocity is change in position over change in time, by extension, average acceleration is change in velocity over change in time. Velocity = v (t) Average acceleration EXAMPLE 5.21 The position of a particle moving along the x -axis is defined by For t ≥ 0, find when the particle is moving right. SOLUTION The particle is moving right when its velocity is positive. Find when the velocity is 0 and test the intervals between those times. 0< t <2 2< t <4 t > 4 v (1) = 6 v (3) = –6 v (5) = 30 The particle is moving right in the time intervals 0 < t < 2 and t > 4. EXAMPLE 5.22 A particle is moving along the x -axis. Figure 5.20 shows its velocity graph. On the interval [0, 6], when is the speed of the particle greatest? Figure 5.20 SOLUTION Since speed is the absolute value of velocity, speed is greatest at about t = 5 seconds, not 2 seconds. 5.5 OPTIMIZATION Optimization is one of the most interesting applications of derivatives. It applies maximizing and minimizing to practical problems, and it involves a wide diversity of problems...
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...
