Question:

# If during a race, a sprinter increases from 5.0 m/s to 7.5 m/s over a period of 1.25 s. What is the sprinter's average acceleration during this period?

## Acceleration is the rate of change of velocity over time. So 7.5 divided by 5.0, find your quotient then divide that number by 1.25 seconds.

In physics, acceleration is the rate at which the velocity of a body changes with time. In general, velocity and acceleration are vector quantities, with magnitude and direction, though in many cases only magnitude is considered (sometimes with negative values for deceleration, treating it as a one dimensional vector). As described by Newton's Second Law, acceleration is caused by a net force; the force, as a vector, is equal to the product of the mass of the object being accelerated (scalar) and the acceleration (vector). The SI unit of acceleration is the meter per second squared (m/s2). For example, an object such as a car that starts from standstill, then travels in a straight line at increasing speed, is accelerating in the direction of travel. If the car changes direction at constant speedometer reading, there is strictly speaking an acceleration although it is often not so described; passengers in the car will experience a force pushing them back into their seats in linear acceleration, and a sideways force on changing direction. If the speed of the car decreases, it is usual and meaningful to speak of deceleration; mathematically it is acceleration in the opposite direction to that of motion. Mathematically, instantaneous acceleration—acceleration over an infinitesimal interval of time—is the rate of change of velocity over time: (Here and elsewhere, if motion is in a straight line, vector quantities can be substituted by scalars in the equations.) Average acceleration over a period of time is the change in velocity $( \Delta \mathbf{v})$ divided by the duration of the period $( \Delta t)$ Acceleration has the dimensions of velocity (L/T) divided by time, i.e., L/T2. The SI unit of acceleration is the metre per second squared (m/s2); this can be called more meaningfully "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second. An object moving in a circular motion—such as a satellite orbiting the earth—is accelerating due to the change of direction of motion, although the magnitude (speed) may be constant. When an object is executing such a motion where it changes direction, but not speed, it is said to be undergoing centripetal (directed towards the center) acceleration. Oppositely, a change in the speed of an object, but not its direction of motion, is a tangential acceleration. Proper acceleration, the acceleration of a body relative to a free-fall condition, is measured by an instrument called an accelerometer. In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e., sum of all forces) acting on it (Newton's second law): where F is the net force acting on the body, m is the mass of the body, and a is the center-of-mass acceleration. As speeds approach the speed of light, relativistic effects become increasingly large and acceleration becomes less. The velocity of a particle moving on a curved path as a function of time can be written as: with v(t) equal to the speed of travel along the path, and a unit vector tangent to the path pointing in the direction of motion at the chosen moment in time. Taking into account both the changing speed v(t) and the changing direction of ut, the acceleration of a particle moving on a curved path can be written using the chain rule of differentiation for the product of two functions of time as: where un is the unit (inward) normal vector to the particle's trajectory (also called the principal normal), and r is its instantaneous radius of curvature based upon the osculating circle at time t. These components are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion, see also circular motion and centripetal force). Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas. Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an equal amount in every equal time period. A frequently cited example of uniform acceleration is that of an object in free fall in a uniform gravitational field. The acceleration of a falling body in the absence of resistances to motion is dependent only on the gravitational field strength g (also called acceleration due to gravity). By Newton's Second Law the force, F, acting on a body is given by: Due to the simple algebraic properties of constant acceleration in the one-dimensional case (that is, the case of acceleration aligned with the initial velocity), there are simple formulas that relate the following quantities: displacement, initial velocity u, final velocity v, acceleration a, and time t: where In the case of uniform acceleration of an object that is initially moving in a direction not aligned with the acceleration, the motion can be resolved into two orthogonal parts, one of constant velocity and the other according to the above equations. As Galileo showed, the net result is parabolic motion, as in the trajectory of a cannonball, neglecting air resistance. Uniform circular motion, that is constant speed along a circular path, is an example of a body experiencing acceleration resulting in velocity of a constant magnitude but change of direction. In this case, because the direction of the object's motion is constantly changing, being tangential to the circle, the object's linear velocity vector also changes, but its speed does not. This acceleration is a radial acceleration since it is always directed toward the centre of the circle and takes the magnitude: where $v$ is the object's linear speed along the circular path. Equivalently, the radial acceleration vector ($\mathbf {a}$) may be calculated from the object's angular velocity $\omega$, whence: where $\mathbf{r}$ is a vector directed from the centre of the circle and equal in magnitude to the radius. The negative shows that the acceleration vector is directed towards the centre of the circle (opposite to the radius). The acceleration, hence also the net force acting on a body in uniform circular motion, is directed toward the centre of the circle; that is, it is centripetal. Whereas the so-called 'centrifugal force' appearing to act outward on the body is really a pseudo force experienced in the frame of reference of the body in circular motion, due to the body's linear momentum at a tangent to the circle. With nonuniform circular motion, i.e., the speed along the curved path changes, a transverse accleration is produced equal to the rate of change of the angular speed around the circle times the radius of the circle. That is, The transverse (or tangential) acceleration is directed at right angles to the radius vector and takes the sign of the angular acceleration ($\alpha$). The special theory of relativity describes the behavior of objects traveling relative to other objects at speeds approaching that of light in a vacuum. Newtonian mechanics is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations. As speeds approach that of light, the acceleration produced by a given force decreases, becoming infinitesimally small as light speed is approached; an object with mass can approach this speed asymptotically, but never reach it. Unless the state of motion of an object is known, it is totally impossible to distinguish whether an observed force is due to gravity or to acceleration—gravity and inertial acceleration have identical effects. Albert Einstein called this the principle of equivalence, and said that only observers who feel no force at all—including the force of gravity—are justified in concluding that they are not accelerating.
Sprints are short running events in athletics and track and field. Races over short distances are among the oldest running competitions. The first 13 editions of the Ancient Olympic Games featured only one event—the stadion race, which was a race from one end of the stadium to the other. There are three sprinting events which are currently held at the Summer Olympics and outdoor World Championships: the 100 metres, 200 metres, and 400 metres. These events have their roots in races of imperial measurements which were later altered to metric: the 100 m evolved from the 100 yard dash, the 200 m distances came from the furlong (or 1/8 of a mile), and the 400 m was the successor to the 440 yard dash or quarter-mile race. At the professional level, sprinters begin the race by assuming a crouching position in the starting blocks before leaning forward and gradually moving into an upright position as the race progresses and momentum is gained. The set position differs depending on the start. Body alignment is of key importance in producing the optimal amount of force. Ideally the athlete should begin in a 4-point stance and push off using both legs for maximum force production. Athletes remain in the same lane on the running track throughout all sprinting events, with the sole exception of the 400 m indoors. Races up to 100 m are largely focused upon acceleration to an athlete's maximum speed. All sprints beyond this distance increasingly incorporate an element of endurance. Human physiology dictates that a runner's near-top speed cannot be maintained for more than 30–35 seconds due to the accumulation of lactic acid in muscles. The 60 metres is a common indoor event and it is an indoor world championship event. Less common events include the 50 metres, 55 metres, 300 metres and 500 metres which are used in some high school and collegiate competitions in the United States. The 150 metres, though rarely competed, has a star-studded history: Pietro Mennea set a world best in 1983, Olympic champions Michael Johnson and Donovan Bailey went head-to-head over the distance in 1997, and Usain Bolt improved Mennea's record in 2009. Biological factors that determine a sprinter's potential include: Note: Indoor distances are less standardized as many facilities run shorter or occasionally longer distances depending on available space. 60m is the championship distance. Starting blocks are used for all competition sprint (up to and including 400 m) and relay events (first leg only, up to 4x400 m). The starting blocks consist of two adjustable footplates attached to a rigid frame. Races commence with the firing of the starter's gun. The starting commands are "On your marks" and "Set". Once all athletes are in the set position, the starter's gun is fired, officially starting the race. For the 100 m, all competitors are lined up side-by-side. For the 200 m, 300 m and 400 m, which involve curves, runners are staggered for the start. In the rare event that there are technical issues with a start, a green card is shown to all the athletes. The green card carries no penalty. If an athlete is unhappy with track conditions after the "on your marks" command is given, the athlete must raise a hand before the "set" command and provide the Start referee with a reason. It is then up to the Start referee to decide if the reason is valid. In the event that the Start referee deems the reason invalid, a yellow card (warning) is issued to that particular athlete. In the event that the athlete is already on a warning the athlete is disqualified. "An athlete, after assuming a full and final set position, shall not commence his(/her) starting motion until after receiving the report of the gun, or approved starting apparatus. If, in the judgement of the Starter or Recallers, he does so any earlier, it shall be deemed a false start." The 100 m Olympic Gold and Silver medallist, Linford Christie of Great Britain famously had frequent false starts that were marginally below the legal reaction time of 0.1 seconds. Christie and his coach, Ron Roddan, both claimed that the false starts were due to Christie's exceptional reaction times being under the legal time. His frequent false starting eventually led to his disqualification from the 1996 Summer Olympics 100 m final in Atlanta, Georgia, USA due to a second false start by Christie. Since January 2010, under IAAF rules, a single false start by an athlete results in disqualification. For all Olympic sprint events, runners must remain within their pre-assigned lanes, which measure 1.22 metres (4 feet) wide, from start to finish. The lanes can be numbered 1 through normally 8 or 9 rarely 10, starting with the inside lane. Any athlete who runs outside the assigned lane to gain an advantage is subject to disqualification. If the athlete is forced to run outside of his or her lane by another person, and no material advantage is gained, there will be no disqualification. Also, a runner who strays from his or her lane in the straightaway, or crosses the outer line of his or her lane on the bend, and gains no advantage by it, will not be disqualified as long as no other runner is obstructed. The first athlete whose torso reaches the vertical plane of the closest edge of the finish line is the winner. To ensure that the sprinter's torso triggers the timing impulse at the finish line rather than an arm, foot, or other body part, a double Photocell is commonly used. Times are only recorded by an electronic timing system when both of these Photocells are simultaneously blocked. Photo finish systems are also used at some track and field events.
Standard gravity, or standard acceleration due to free fall, usually denoted by $g_0$ or $g_n$, is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about (≈ or ≈). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from rotation of the earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5 percent greater at the poles than at the equator. Although the symbol $g$ is sometimes used for standard gravity, $g$ (without a suffix) can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth (see Earth's gravity). The symbol $g$ should not be confused with $G$, the gravitational constant, or g, the symbol for gram. The $g$ is also used as a unit for any form of acceleration, with the value defined as above; see g-force. The value of $g_0$ defined above is a nominal midrange value on Earth, originally based on the acceleration of a body in free fall at sea level at a geodetic latitude of 45°. Although the actual acceleration of free fall on Earth varies according to location, the above standard figure is always used for metrological purposes. (The actual average sea-level acceleration on Earth is slightly less.)
A cycling sprinter is a road bicycle racer or track racer who can finish a race very explosively by accelerating quickly to a high speed, often using the slipstream of another cyclist or group of cyclists tactically to conserve energy. Apart from using sprint as a racing tactic, sprinters can also compete for intermediate sprints (sometimes called primes), often to provide additional excitement in cities along the route of a race. In stage races, intermediate sprints and final stage placings may be combined in a points classification. For example, in the points classification in the Tour de France, the maillot vert (green jersey) is won by the race's most consistent sprinter. At the Tour de France, the most successful competitor for this honor is German sprinter Erik Zabel, who won a record six Tour de France green jerseys (1996–2001). Sprinters have a higher ratio of fast-twitch muscle fibers than non-sprinters. Road cycling sprinters sometimes tend to have a larger build than the average road racing cyclist, combining the strength of their legs with their upper body to produce a short burst of speed necessary in a closely contested finish. Some sprinters have a high top speed but may take a longer distance to achieve it, while others can produce short and sharp accelerations. A sprinter is usually heavier, limiting their speed advantage to relatively flat sections. It is therefore not uncommon for sprinters to be dropped by the peloton (also known as the 'bunch' or 'pack') if a race is through hilly terrain. Sprinters may have different preferences. Some prefer a longer "launch" while others prefer to 'draft' or slipstream behind their team-mates or opponents before accelerating in the final meters. Some prefer slight uphill finishes, while others prefer downhill finishes. In conventional road races, sprinters may bide their time waiting until the last few hundred metres before putting on a burst of speed to win the race. Many races will finish with a large group sprinting for the win; some sprinters may have team-mates, so-called domestiques 'leading them out' (i.e., keeping pace high and sheltering the sprinter) so that they have a greater chance of finishing in the leading positions. These team-mates tend to "peel off" one by one as they tire; the last team-mate is known as the "lead-out sprinter" and the best of them are excellent sprinters in their own right. Several of the Classic one day races, for example Milan-San Remo or Paris–Tours tend to favour sprinters because of their long distance and relatively flat terrain. Most editions of these races end in a bunch sprint, often won by racers also successful in the points classification at stage races. For example, Zabel has won Milan–San Remo four times and Paris–Tours three times. Stronger sprinters with abilities in hilly terrain or on cobblestones also have good prospects of winning other major classics such as the Tour of Flanders or the Amstel Gold Race. Successful sprinters of the past include Freddy Maertens, Mario Cipollini and Erik Zabel. Cipollini holds the record for most stage wins in the Grand Tours as a sprinter; 57, of which 42 in the Giro d'Italia. Zabel won 41 stages, and 10 Points Classifications in both Giro, Tour (six times) and Vuelta (three times). Like Zabel, Italian sprinter Alessandro Petacchi won stages and the Points Classification of all three grand tours, including 20 stage wins in Spain. The record for stage wins in the Vuelta belongs to Delio Rodriguez with 39 wins. Mark Cavendish was named the Tour de France's best sprinter of all time by French paper L'Equipe on July 15, 2012. A good sprint can also secure a lot of victories for other specialists, such as Classics riders and GC-contenders. Seán Kelly won 21 stages in the Tour and Vuelta, and the Points Classifications of both races four times each, in addition to his nine major Classics wins. Likewise, Belgian classics specialists Rik van Looy and Roger de Vlaeminck were very successful due to a good final sprint, as was - more recently - two times World Champion Paolo Bettini. Conversely, many sprinters put to use their abilities to win more than just stages, and were successful in classics such as the Tour of Flanders (like Jan Raas) and the Giro di Lombardia (like André Darrigade). The ultimate sprinter classic due to its relatively flat course is Milan – San Remo, won four times by Zabel and three times by three-time World Champion Oscar Freire. Other "flat" one day races considered important sprinters classics include Scheldeprijs, Vattenfall Cyclassics and Paris-Tours. Sprinting on a cycle track or velodrome ranges from the highly specialised sprint event (where two - sometimes three or more - riders slowly circle the track looking to gain a tactical advantage before launching a finishing burst over the final 200 metres, which is timed), to massed-start events decided by the first across the line after a certain number of laps (similar to road racing). The sprint specialist may also ride short track time trials over 1000 metres, the team sprint and Keirin events. In Madison racing, a team may comprise a specialist sprinter, for when sudden bursts of speed are required, and another rider able to ride at a more consistent high tempo. The Complete Cycle Sport Guide, Peter Konopka, 1982, EP Publishing
The primary vehicle of calculus and other higher mathematics is the function. Its "input value" is its argument, usually a point ("P") expressible on a graph. The difference between two points, themselves, is known as their Delta (ΔP), as is the difference in their function result, the particular notation being determined by the direction of formation: The general preference is the forward orientation, as F(P) is the base, to which differences (i.e., "ΔP"s) are added to it. Furthermore, The function difference divided by the point difference is known as the difference quotient (attributed to Isaac Newton,][ it is also known as Newton's quotient): If ΔP is infinitesimal, then the difference quotient is a derivative, otherwise it is a divided difference: Regardless if ΔP is infinitesimal or finite, there is (at least—in the case of the derivative—theoretically) a point range, where the boundaries are P ± (.5)ΔP (depending on the orientation—ΔF(P), δF(P) or ∇F(P)): Derivatives can be regarded as functions themselves, harboring their own derivatives. Thus each function is home to sequential degrees ("higher orders") of derivation, or differentiation. This property can be generalized to all difference quotients.
As this sequencing requires a corresponding boundary splintering, it is practical to break up the point range into smaller, equi-sized sections, with each section being marked by an intermediary point ("Pi"), where LB = P0 and UB = Pń, the nth point, equaling the degree/order: There are other derivative notations, but these are the most recognized, standard designations. The quintessential application of the divided difference is in the presentation of the definite integral, which is nothing more than a finite difference: Given that the mean value, derivative expression form provides all of the same information as the classical integral notation, the mean value form may be the preferable expression, such as in writing venues that only support/accept standard ASCII text, or in cases that only require the average derivative (such as when finding the average radius in an elliptic integral). This is especially true for definite integrals that technically have (e.g.) 0 and either $\pi\,\!$ or $2\pi\,\!$ as boundaries, with the same divided difference found as that with boundaries of 0 and $\begin{matrix}\frac{\pi}{2}\end{matrix}$ (thus requiring less averaging effort): This also becomes particularly useful when dealing with iterated and smultiple integral (ΔA = AU - AL, ΔB = BU - BL, ΔC = CU - CL): Hence, and
In kinematics, velocity is the rate of change of the position of an object, equivalent to a specification of its speed and direction of motion. For motion in one dimension, velocity can be defined as the slope of the position vs. time graph of an object. Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path (the object's path does not curve). Thus, a constant velocity means motion in a straight line at a constant speed. If there is a change in speed, direction, or both, then the object is said to have a changing velocity and is undergoing an acceleration. For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration. Velocity is a vector physical quantity; both magnitude and direction are required to define it. The scalar absolute value (magnitude) of velocity is called "speed", a quantity that is measured in metres per second (m/s or m⋅s−1) when using the SI (metric) system. For example, "5 metres per second" is a scalar (not a vector), whereas "5 metres per second east" is a vector. The rate of change of velocity (in m/s) as a function of time (in s) is "acceleration" (in m/s2 – stated "metres per second per second"), which describes how an object's speed and direction of travel change at each point in time. The average velocity $\boldsymbol{\bar{v}}$ of an object moving through a displacement $( \Delta \boldsymbol{x})$ during a time interval $( \Delta t)$ is described by the formula: The velocity vector v of an object that has positions x(t) at time t and x$(t + \Delta t)$ at time $t + \Delta t$, can be computed as the derivative of position: Velocity is also defined as rate of change of displacement. Average velocity magnitudes are always smaller than or equal to average speed of a given particle. Instantaneous velocity is always tangential to trajectory. Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity. The equation for an object's velocity can be obtained mathematically by evaluating the integral of the equation for its acceleration beginning from some initial period time $t_0$ to some point in time later $t_n$. The final velocity v of an object which starts with velocity u and then accelerates at constant acceleration a for a period of time $\Delta t$ is: The average velocity of an object undergoing constant acceleration is $\tfrac {(\boldsymbol{u} + \boldsymbol{v})}{2}$, where u is the initial velocity and v is the final velocity. To find the position, x, of such an accelerating object during a time interval, $\Delta t$, then: When only the object's initial velocity is known, the expression, can be used. This can be expanded to give the position at any time t in the following way: These basic equations for final velocity and position can be combined to form an equation that is independent of time, also known as Torricelli's equation: The above equations are valid for both Newtonian mechanics and special relativity. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words only relative velocity can be calculated. In Newtonian mechanics, the kinetic energy (energy of motion), $E_K$, of a moving object is linear with both its mass and the square of its velocity: The kinetic energy is a scalar quantity. Escape velocity is the minimum a ballistic object needs to escape from a massive body like the earth. It represents the kinetic energy that when added to the object's gravitational potential energy (which is always negative) is greater than or equal to zero. Escape velocity from the Earth's surface is about 11 100 m/s. Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame. If an object A is moving with velocity vector v and an object B with velocity vector w, then the velocity of object A relative to object B is defined as the difference of the two velocity vectors: Similarly the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is: Usually the inertial frame is chosen in which the latter of the two mentioned objects is in rest. In the one dimensional case, the velocities are scalars and the equation is either: In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and an angular velocity, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). The radial and angular velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity is the component of velocity along a circle centered at the origin. where The magnitude of the radial velocity is the dot product of the velocity vector and the unit vector in the direction of the displacement. where The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. It is also the product of the angular speed $\omega$ and the magnitude of the displacement. such that Angular momentum in scalar form is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity. where The expression $mr^2$ is known as moment of inertia. If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion.
Metre per second (U.S. spelling: meter per second) is an SI derived unit of both speed (scalar) and velocity (vector quantity which specifies both magnitude and a specific direction), defined by distance in metres divided by time in seconds. The SI unit symbols are m·s−1, m s−1, m/s, or . Where metres per second are several orders of magnitude too slow to be convenient, such as in astronomical measurements, velocities may be given in kilometres per second, where 1 km/s is 1000 metres per second. 1 m/s is equivalent to: 1 foot per second = 0.3048 m·s−1 (exactly) 1 mile per hour = 0.44704 m·s−1 (exactly) 1 −1km·h = 0.2 m·s−1 (exactly) 1 kilometre per second is equivalent to: Although m·s−1 is an SI derived unit, it could be viewed as more fundamental than the metre, since the metre is now derived from the speed of light in vacuum, which is defined as exactly 299 792 458 m·s−1 by the BIPM. One metre per second is 1/299 792 458 of the speed of light in vacuum. The benz, named in honour of Karl Benz, has been proposed as a name for one metre per second. Although it has seen some support as a practical unit, primarily from German sources, it was rejected as the SI unit of velocity and has not seen widespread use or acceptance.
sprinter Sports Physical quantities Dynamics Acceleration Sprint Velocity Physics Kinematics Motion

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