# 3.3 Average and instantaneous acceleration  (Page 2/7)

 Page 2 / 7

The term deceleration can cause confusion in our analysis because it is not a vector and it does not point to a specific direction with respect to a coordinate system, so we do not use it. Acceleration is a vector, so we must choose the appropriate sign for it in our chosen coordinate system. In the case of the train in [link] , acceleration is in the negative direction in the chosen coordinate system , so we say the train is undergoing negative acceleration.

If an object in motion has a velocity in the positive direction with respect to a chosen origin and it acquires a constant negative acceleration, the object eventually comes to a rest and reverses direction. If we wait long enough, the object passes through the origin going in the opposite direction. This is illustrated in [link] .

## Calculating average acceleration: a racehorse leaves the gate

A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration?

## Strategy

First we draw a sketch and assign a coordinate system to the problem [link] . This is a simple problem, but it always helps to visualize it. Notice that we assign east as positive and west as negative. Thus, in this case, we have negative velocity.

We can solve this problem by identifying $\text{Δ}v\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{Δ}t$ from the given information, and then calculating the average acceleration directly from the equation $\stackrel{\text{–}}{a}=\frac{\text{Δ}v}{\text{Δ}t}=\frac{{v}_{\text{f}}-{v}_{0}}{{t}_{\text{f}}-{t}_{0}}$ .

## Solution

First, identify the knowns: ${v}_{0}=0,{v}_{\text{f}}=-15.0\phantom{\rule{0.2em}{0ex}}\text{m/s}$ (the negative sign indicates direction toward the west), Δ t = 1.80 s.

Second, find the change in velocity. Since the horse is going from zero to –15.0 m/s, its change in velocity equals its final velocity:

$\text{Δ}v={v}_{\text{f}}-{v}_{0}={v}_{\text{f}}=-15.0\phantom{\rule{0.2em}{0ex}}\text{m/s}.$

Last, substitute the known values ( $\text{Δ}v\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{Δ}t$ ) and solve for the unknown $\stackrel{\text{–}}{a}$ :

$\stackrel{\text{–}}{a}=\frac{\text{Δ}v}{\text{Δ}t}=\frac{-15.0\phantom{\rule{0.2em}{0ex}}\text{m/s}}{1.80\phantom{\rule{0.2em}{0ex}}\text{s}}=-8.33{\text{m/s}}^{2}.$

## Significance

The negative sign for acceleration indicates that acceleration is toward the west. An acceleration of 8.33 m/s 2 due west means the horse increases its velocity by 8.33 m/s due west each second; that is, 8.33 meters per second per second, which we write as 8.33 m/s 2 . This is truly an average acceleration, because the ride is not smooth. We see later that an acceleration of this magnitude would require the rider to hang on with a force nearly equal to his weight.

Check Your Understanding Protons in a linear accelerator are accelerated from rest to $2.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{m/s}$ in 10 –4 s. What is the average acceleration of the protons?

Inserting the knowns, we have
$\stackrel{\text{–}}{a}=\frac{\text{Δ}v}{\text{Δ}t}=\frac{2.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{m/s}-0}{{10}^{-4}\phantom{\rule{0.2em}{0ex}}\text{s}-0}=2.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{11}{\text{m/s}}^{2}.$

## Instantaneous acceleration

Instantaneous acceleration a , or acceleration at a specific instant in time , is obtained using the same process discussed for instantaneous velocity. That is, we calculate the average velocity between two points in time separated by $\text{Δ}t$ and let $\text{Δ}t$ approach zero. The result is the derivative of the velocity function v ( t ), which is instantaneous acceleration    and is expressed mathematically as

how can I become confident in physics
practice more and more question ,,,,if u get trouble in any one question infact do that again and again.
jyotirmayee
Can someone please tell what really happened to planet Pluto
what are the first 20 elements in periodic table.
H,O...
Arzoodan
what happened to the rest now, or have you forgotten?
Ejiba
who discover periodic table?
it wasn't discovered , it was made.and the person who made it was dmitri mendeleev. dobreinier and newland gave their laws before dmitri related periodic table but wasn't successful in their work
Ritik
Nope, numerous number of scientist had actually contributed in the making of periodic table. Dmitri Mendeleev succeeded making all the elements into the right order in accordance to their atomic number.
Dame
what is the Greek name for calcium
Greek word for calcium is asvestio while Latin name is calf meaning lime or limestone
Ejiba
different types of wave
longitudinal and transverse waves
Ravindra
a gun is kept in the state that it cannot move anywhere and the bullet is fired. Then what is the effect on the velocity of bullet and KE of gun ?
want is meant by the term solar system
it refers to the sun and all heavenly bodies revolving around it.
Danie
excatly...for sure
Arzoodan
in addition to Danie's, a solar system is a collection of planets and their moons, asteroids, and other objects bound together by the Star's gravitational force directly or indirectly.
Galiwango
what is meant by total internal reflection
what iw meant by total internal reflection
Akshay
Lorentz force?
jyotirmayee
study fibre optics. .you will get total internal reflection
Siddhansh
wha
jyotirmayee
what is Lorentz force?
jyotirmayee
a ray of light traveling at an angle of incidence greater than critical angle from denser to rarer medium is totally reflected back into the denser medium is called total internal reflaction
Manoj
motion in strat line where is this chapter
motion in straight line is kinematic's part
Ritik
yea
Manoj
this defination isn't correct
Arzoodan
motion in one dimension
Anil
what is Lorentz force?
what is maxwell electromagnetic law?
jyotirmayee
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
Akshay
what's the working difference between a dynamo and a pump?
a dynamo is basically a dc generator while pump is usually equipped with a motor
vedanth
a dynamo converts mechanical energy to electrical while a pump is opposite to that
vedanth
nice
Piyali
okay
Friday
why sea water looks bluish?
cuz the sky is blue...
Mehdi
you see the reflexion of the "blue" sky in the water
Mehdi
somewhere sea water turns green why?
Piyali
never seen bro... are u sure ?
Mehdi
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
Piyali
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Mehdi
because of the reflection of the sky
Friday
rays coming from the sun consist of all 7 colours ie.VIBGYOR. when the ray strikes surface of water,all colors gets absorbed by it except blue which gets reflected by it.so we find the sea water appearing bluish
Ritik
how can someone identify sea water from rain water
Oniyide
i think it's not possible as because rainwater consists of water from all kind of water bodies ie.lakes,seas etc but u can predict if u have a sea nearby ur home or city
Ritik
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me too
Nirupam
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Nirupam
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john
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jyotirmayee
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jyotirmayee
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Nirupam
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Nirupam
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jyotirmayee
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Nirupam
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jyotirmayee