what is acceleration formula

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what is acceleration formula

definition

When the velocity of an object changes it is said to be accelerating. Acceleration is the rate of change of velocity with time.

what is acceleration formula: In everyday English, the word acceleration is often used to describe a state of increasing speed. For many Americans, their only experience with acceleration comes from car ads. When a commercial shouts “zero to sixty in six points seven seconds” what they’re saying here is that this particular car takes 6.7 s to reach a speed of 60 mph starting from a complete stop. This example illustrates acceleration as it is commonly understood, but acceleration in physics is much more than just increasing speed.

Any change in the velocity of an object results in an acceleration: increasing speed (what people usually mean when they say acceleration), decreasing speed (also called deceleration or retardation), or changing direction (called centripetal acceleration). Yes, that’s right, a change in the direction of motion results in an acceleration even if the moving object neither sped up nor slowed down. That’s because acceleration depends on the change in velocity and velocity is a vector quantity — one with both magnitude and direction. Thus, a falling apple accelerates, a car stopping at a traffic light accelerates, and an orbiting planet accelerates. Acceleration occurs anytime an object’s speed increases or decreases, or it changes direction.

Much like velocity, there are two kinds of acceleration: average and instantaneous. Average acceleration is determined over a “long” time interval. The word long in this context means finite — something with a beginning and an end. The velocity at the beginning of this interval is called the initial velocity, represented by the symbol v0 (vee nought), and the velocity at the end is called the final velocity, represented by the symbol v (vee). Average acceleration is a quantity calculated from two velocity measurements.

 =Δv =v − v0
ΔtΔt

In contrast, instantaneous acceleration is measured over a “short” time interval. The word short in this context means infinitely small or infinitesimal — having no duration or extent whatsoever. It’s a mathematical ideal that can can only be realized as a limit. The limit of a rate as the denominator approaches zero is called a derivative. Instantaneous acceleration is then the limit of average acceleration as the time interval approaches zero — or alternatively, acceleration is the derivative of velocity.

a =

lim

Δv

 =

dv

Δt→0

Δt

dt

Acceleration is the derivative of velocity with time, but velocity is itself the derivative of displacement with time. The derivative is a mathematical operation that can be applied multiple times to a pair of changing quantities. Doing it once gives you a first derivative. Doing it twice (the derivative of a derivative) gives you a second derivative. That makes acceleration the first derivative of velocity with time and the second derivative of displacement with time.

a =

dv

 =

d

ds

 =

d2s

dt

dt

dt

dt2

A word about notation. In formal mathematical writing, vectors are written in boldface. Scalars and the magnitudes of vectors are written in italics. Numbers, measurements, and units are written in roman (not italic, not bold, not oblique — ordinary text). For example…

a = 9.8 m/s2, θ = −90°

 or

a = 9.8 m/s2 at −90°

(Design note: I think Greek letters don’t look good on the screen when italicized so I have decided to ignore this rule for Greek letters until good looking Greek fonts are the norm on the web.)

units

Calculating acceleration involves dividing velocity by time — or in terms of units, dividing meters per second [m/s] by second [s]. Dividing distance by time twice is the same as dividing distance by the square of time. Thus the SI unit of acceleration is the meter per second squared.

m

 =

m/s

 =

m

1

s2

s

ss

Another frequently used unit is the acceleration due to gravity — g. Since we are all familiar with the effects of gravity on ourselves and the objects around us it makes for a convenient standard for comparing accelerations. Everything feels normal at 1 g, twice as heavy at 2 g, and weightless at 0 g. This unit has a precisely defined value of 9.80665 m/s2, but for everyday use 9.8 m/s2 is sufficient, and 10 m/s2 is convenient for quick estimates.

The unit called acceleration due to gravity (represented by a roman g) is not the same as the natural phenomena called acceleration due to gravity (represented by an italic g). The former has a defined value whereas the latter has to be measured. (More on this later.)

Although the term “g force” is often used, the g is a measure of acceleration, not force. (More on forces later.) Of particular concern to humans are the physiological effects of acceleration. To put things in perspective, all values are stated in g.

  • In roller coaster design, speed is of the essence. Or, is it? If speed was all there was to designing a thrill ride, then the freeway would be pretty exciting. Most roller coaster rarely exceed 30 m/s (60 mph). Contrary to popular belief, it is the acceleration that makes the ride interesting. A well designed roller coaster will subject the rider to maximum accelerations of 3 to 4 g for brief periods. This is what gives the ride its dangerous feel.
  • Despite the immense power of its engines, the acceleration of the Space Shuttle is kept below 3 g. Anything greater would put unnecessary stress on the astronauts, the payload, and the ship itself. Once in orbit, the whole system enters into an extended period of free fall, which provides the sensation of weightlessness. Such a “zero g” environment can also be simulated inside a specially piloted aircraft or a free fall drop tower. (More on this later.)
  • Fighter pilots can experience accelerations of up to 8 g for brief periods during tactical maneuvers. If sustained for more than a few seconds, 4 to 6 g is sufficient to induce blackout. To prevent “g-force loss of consciousness” (G-LOC), fighter pilots wear special pressure suits that squeeze the legs and abdomen, forcing blood to remain in the head.
  • Pilots and astronauts may also train in human centrifuges capable of up to 15 g. Exposure to such intense accelerations is kept very brief for safety reasons. Humans are rarely subjected to anything higher than 8 g for longer than a few seconds.
  • Acceleration is related to injury. This is why the most common sensor in a crash test dummy is the accelerometer. Extreme acceleration can lead to death. The acceleration during the crash that killed Diana, Princess of Wales, in 1997 was estimated to have been on the order of 70 to 100 g, which was intense enough to tear the pulmonary artery from her heart — an injury that is nearly impossible to survive. Had she been wearing a seat belt, the acceleration would have been something more like 30 or 35 g — enough to break a rib or two, but not nearly enough to kill most people.

Here are some sample accelerations to end this section.

a (m/s2)

event

5 × 10−14

smallest acceleration in a scientific experiment

2 × 10−10

galactic acceleration at the sun

9 × 10−10

anomalous acceleration of

pioneer

spacecraft

0.5

elevator, hydraulic

0.63

free fall

acceleration on

pluto

1

elevator, cable

1.6

free fall

acceleration on the moon

8.8

International Space Station

3.7

free fall

acceleration on mars

9.8

free fall

acceleration on earth

10–40

manned rocket at launch

20

space shuttle, peak

24.8

free fall

acceleration on

jupiter

20–50

roller coaster

80

limit of sustained human tolerance

0–150

human training centrifuge

100–200

ejection seat

270

free fall

acceleration on the sun

600

airbags automatically deploy

104–106

medical centrifuge

106

bullet in the barrel of a gun

106

free fall

acceleration on a white dwarf star

1012

free fall

acceleration on a neutron star

Acceleration of selected events (smallest to largest)

event

typical car

sports car

F-1 race car

large truck

starting

0.3–0.5

0.5–0.9

1.7

< 0.2

braking

0.8–1.0

1.0–1.3

2

~ 0.6

cornering

0.7–0.9

0.9–1.0

3

?

Automotive accelerations (g)

 

a (g)

event

2.9

sneeze

3.5

cough

3.6

crowd jostle

4.1

slap on

back

8.1

hop off

step

10.1

plop down in

chair

60

chest acceleration during

car

crash at 48 km/h with airbag

70–100

crash that killed Diana, Princess of Wales, 1997

150–200

head acceleration limit during bicycle crash with

helmet

Acceleration and the human body

 

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