Acceleration and Velocity
Acceleration is the rate of change of velocity over time, indicating how quickly an object's speed is changing. Velocity, on the other hand, is the rate of change of an object's position with respect to time, including both speed and direction. In mathematical terms, acceleration is the derivative of velocity with respect to time.
7 Key excerpts on "Acceleration and 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)
...Angular motion involves movement around an axis of rotation. Scalar quantity A quantity that is represented by magnitude (size) only. Vector quantity A quantity that is represented by both magnitude and direction. Distance and displacement The term distance is classified as a scalar quantity and is expressed with reference to magnitude only (e.g. 140 mi (miles)). Displacement is the vector quantity and is expressed with both magnitude and direction (e.g. 100 m along a track in a straight line from point A to point B). Speed and velocity Speed is the scalar quantity that is used to describe the motion of an object (e.g. 4 m/s). It is calculated as distance divided by time taken. Velocity is the vector quantity and it is also used to describe the motion of an object (e.g. 4.5 m/s from north to south). It is calculated as displacement divided by time taken. Acceleration Is defined as the change in velocity per unit of time and hence is a vector quantity. It is calculated as velocity divided by time taken. Average and instantaneous Average is the usual term for the arithmetic mean (in this context often over larger periods of time or displacement). Instantaneous (or tending to instantaneous (towards zero)) refers to smaller increments of time or displacement in which the velocity or acceleration calculations are made. The smaller the increments of time between successive data points the more the value tends towards an instantaneous value. FURTHER READING The following two resources provide additional reading on the concept of linear kinematics in sports and exercise. 1 McDonald, C., & Dapena, J. (1991). Linear kinematics of the men’s 110-m and women’s 100-m hurdles races. Medicine and Science in Sports and Exercise, 23 (12), 1382–1391. 2 Murphy, A. J., Lockie, R. G., & Coutts, A. J. (2003)...
eBook - ePub- W. Bolton(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...Acceleration is the rate at which the velocity changes. Thus, for the graph shown in Figure 4.9, the velocity changes from v 1 to v 2 when the time changes from t 1 to t 2. Thus the acceleration over that time interval is (v 2 - v 1)/(t 2 - t 1). But this is the gradient of the graph. Thus: acceleration = gradient of the velocity–time graph With a straight- line graph, the gradient is the same for all points and so we have a uniform acceleration. When the graph is not a straight line, as in Figure 4.10, the acceleration is no longer uniform. (v 2 - v 1) is the change in velocity in a time of (t 2 - t 1) and thus (v 2 - v 1)/(t 2 - t 1) represents the average acceleration over that time. The smaller we make the times between A and B then the more the average is taken over a smaller time interval and more closely approximates to the instantaneous acceleration. An infinitesimally small time interval means we have the tangent to the curve. Thus if we want the acceleration at an instant of time then we have to determine the gradient of the tangent to the graph at that time, i.e. instantaneous acceleration = gradient of tangent to the velocity–time graph at that instant Figure 4.10 Velocity–time graph The distance travelled by an object in a particular time interval is average velocity over a time interval × the time interval. Thus if the velocity changes from v 1 at time t 1 to v 2 at time t 2, as in Figure 4.11, then the distance travelled between t 1 and t 2 is represented by the product of the average velocity and the time interval. But this is equal to the area under the graph line between t 1 and t 2. Thus: distance travelled between t 1 and t 2 = area under the graph between these times Figure 4.11 Distance travelled Example Figure 4.12 shows the velocity–time graph for a train travelling between two stations. What is (a) the distance between the stations and (b) the initial acceleration? Figure 4.12 Example The distance travelled is the area under the graph...
eBook - ePub- A. D. Johnson(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...Motion of bodies 2 2.1 Aims To introduce the concepts of linear and angular motion. To explain the relationships between displacement, velocity and acceleration. To explain the relationship between absolute and relative velocities. To define the equations used to analyse linear and angular motion. To introduce an approach by which linear and angular motion problems can be analysed. To explain related topics such as ‘falling bodies’, ‘trajectories’ and vector methods. 2.2 Introduction to Motion When traffic lights turn to green a car will move away with increasing velocity. The car will cover a distance in a particular direction and will possess a particular velocity at any instant. During this process the car possesses the three basic constituents of motion, namely: displacement, velocity and acceleration. It should be noted that since the car runs on wheels, these will also be in motion and therefore possess displacement, velocity and acceleration. However, the car moves in a linear direction, while the wheels move in an angular direction. 2.2.1 Displacement If a man walks 10 km, there is an indication of the distance between the start position and the final position, but there is no indication of the direction. The 10 km is merely the distance covered and, as such, is a scalar quantity, i.e. possessing magnitude only. Displacement, however, implies a change in position or movement over a distance and gives the position and direction from the start point. Thus displacement is a vector quantity possessing both magnitude and direction. Fig. 2.1 Displacement diagram. Figure 2.1 gives an example of a man who walks 3 km east then 4 km north. He has actually walked a distance of 7 km but has been displaced from his start point by only 5 km. 2.2.2 Velocity Velocity is the value of displacement measured over a period of time. It is the rate over which a distance/displacement is traversed...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Naturally, velocity is denoted by v (t), and its derivative, acceleration, by a (t). As a particle moves to the right, the x-coordinate is increasing as time is increasing, so the ratio for average velocity, is positive. As the time interval shortens, the instantaneous velocity, is also defined as positive. Similarly, motion to the left defines negative velocity. Often, over a time interval, a particle may move right and left. Although this motion has little impact on instantaneous velocity or acceleration, it has significant implications for average velocity. Average velocity depends on displacement, and displacement is defined simply as final position minus initial position. KEY CONCEPTS OF HORIZONTAL PARTICLE MOTION Displacement Let x (t) be the position of a particle moving along the x -axis. Position is a function of time, t. On a given time interval [ a, b ] the particle’s, displacement = ∆ x = x (b) – x (a). Average Velocity If x (t) is the position of a particle moving along the x -axis, on a given time interval [ a, b ], its average velocity Instantaneous Velocity If x (t) is the position of a particle moving along the x -axis, instantaneous velocity The instantaneous velocity of the particle measured at any moment t = c is v (c) = x ′(c). Speed Speed is the absolute value of velocity. 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)...
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 - 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- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Typical vector quantities are velocity, acceleration and force. Thus, a velocity of 30 km/h due west, and an acceleration of 7 m/s 2 acting vertically downwards, are both vector quantities. A vector quantity is represented by a straight line lying along the line of action of the quantity, and having a length that is proportional to the size of the quantity, as shown in chapter 19. Thus ab in Figure 23.2 represents a velocity of 20 m/s, whose line of action is due west. The bold letters ab indicate a vector quantity and the order of the letters indicate that the line of action is from a to b. Figure 23.2 Consider two aircraft A and B flying at a constant altitude, A travelling due north at 200 m/s and B travelling 30° east of north, written N 30° E, at 300 m/s, as shown in Figure 23.3. Figure 23.3 Relative to a fixed point 0, 0 a represents the velocity of A and 0 b the velocity of B. The velocity of B relative to A, that is the velocity at which B seems to be travelling to an observer on A, is given by ab, and by measurement is 160 m/s in a direction E 22° N. The velocity of A relative to B, that is, the velocity at which A seems to be travelling to an observer on B, isgivenby ba and by measurement is 160 m/s in a direction W 22° S. Problem 8. Two cars are travelling on horizontal roads in straight lines, car A at 70 km/h at N 10° E and car B at 50 km/h at W 60° N. Determine, by drawing a vector diagram to scale, the velocity of car A relative to car B With reference to Figure 23.4 (a), 0 a represents the velocity of car A relative to a fixed point 0, and 0 b represents the velocity of car B relative to a fixed point 0. The velocity of car A relative to car B is given by ba and by measurement is 45 km/h in a direction of E 35° N. Figure 23.4 Problem 9. Verify the result obtained in Problem 8 by calculation The triangle shown in Figure 23.4 (b) is similar to the vector diagram shown in Figure 23.4 (a). Angle B0A is 40°...
