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a ( t ) = d d t v ( t ) .

Thus, similar to velocity being the derivative of the position function, instantaneous acceleration is the derivative of the velocity function. We can show this graphically in the same way as instantaneous velocity. In [link] , instantaneous acceleration at time t 0 is the slope of the tangent line to the velocity-versus-time graph at time t 0 . We see that average acceleration a = Δ v Δ t approaches instantaneous acceleration as Δ t approaches zero. Also in part (a) of the figure, we see that velocity has a maximum when its slope is zero. This time corresponds to the zero of the acceleration function. In part (b), instantaneous acceleration at the minimum velocity is shown, which is also zero, since the slope of the curve is zero there, too. Thus, for a given velocity function, the zeros of the acceleration function give either the minimum or the maximum velocity.

Graph A shows velocity plotted versus time. Velocity increases from t1 to t2 and t3. It reaches maximum at t0. It decreases to t4 and continues to decrease to t5 and t6. The slope of the tangent line at t0 is indicated as the instantaneous velocity. Graph B shows velocity plotted versus time. Velocity decreases from t1 to t2 and t3. It reaches minimum at t0. It increases to t4 and continues to increase to t5 and t6. The slope of the tangent line at t0 is indicated as the instantaneous velocity.
In a graph of velocity versus time, instantaneous acceleration is the slope of the tangent line. (a) Shown is average acceleration a = Δ v Δ t = v f v i t f t i between times Δ t = t 6 t 1 , Δ t = t 5 t 2 , and Δ t = t 4 t 3 . When Δ t 0 , the average acceleration approaches instantaneous acceleration at time t 0. In view (a), instantaneous acceleration is shown for the point on the velocity curve at maximum velocity. At this point, instantaneous acceleration is the slope of the tangent line, which is zero. At any other time, the slope of the tangent line—and thus instantaneous acceleration—would not be zero. (b) Same as (a) but shown for instantaneous acceleration at minimum velocity.

To illustrate this concept, let’s look at two examples. First, a simple example is shown using [link] (b), the velocity-versus-time graph of [link] , to find acceleration graphically. This graph is depicted in [link] (a), which is a straight line. The corresponding graph of acceleration versus time is found from the slope of velocity and is shown in [link] (b). In this example, the velocity function is a straight line with a constant slope, thus acceleration is a constant. In the next example, the velocity function is has a more complicated functional dependence on time.

Graph A shows velocity in meters per second plotted versus time in seconds. Graph is linear and has a negative constant slope. Graph B shows acceleration in meters per second square plotted versus time in seconds. Graph is linear and has a zero slope with the acceleration being equal to -6.
(a, b) The velocity-versus-time graph is linear and has a negative constant slope (a) that is equal to acceleration, shown in (b).

If we know the functional form of velocity, v ( t ), we can calculate instantaneous acceleration a ( t ) at any time point in the motion using [link] .

Calculating instantaneous acceleration

A particle is in motion and is accelerating. The functional form of the velocity is v ( t ) = 20 t 5 t 2 m/s .

  1. Find the functional form of the acceleration.
  2. Find the instantaneous velocity at t = 1, 2, 3, and 5 s.
  3. Find the instantaneous acceleration at t = 1, 2, 3, and 5 s.
  4. Interpret the results of (c) in terms of the directions of the acceleration and velocity vectors.

Strategy

We find the functional form of acceleration by taking the derivative of the velocity function. Then, we calculate the values of instantaneous velocity and acceleration from the given functions for each. For part (d), we need to compare the directions of velocity and acceleration at each time.

Questions & Answers

how can I become confident in physics
arshdeep Reply
practice more and more question ,,,,if u get trouble in any one question infact do that again and again.
jyotirmayee
is theory important or numericals
arshdeep
both ,,,,because without understanding theory ,,,,u can't able to solve problemsss or numericals
jyotirmayee
which books I can follow for IIT jee prep for mcq and which for mcq
arshdeep
Can someone please tell what really happened to planet Pluto
Ejiba Reply
what are the first 20 elements in periodic table.
Ejiba Reply
H,O...
Arzoodan
what happened to the rest now, or have you forgotten?
Ejiba
who discover periodic table?
lovet Reply
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
Oniyide Reply
Greek word for calcium is asvestio while Latin name is calf meaning lime or limestone
Ejiba
different types of wave
Yog Reply
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 ?
Gobinda Reply
want is meant by the term solar system
jafar Reply
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
Akshay Reply
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
Vijaybhai Reply
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?
jyotirmayee Reply
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 Reply
what answer fir this
Akshay
what's the working difference between a dynamo and a pump?
Piyali Reply
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?
Piyali Reply
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
waaaww... you re awesome
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
i need solutions of unuversity pbysucs volume 1
Vimla Reply
me too
Nirupam
Help us if anyone knows
Nirupam
bring questions
john
actually if u wanted whole book solution then u should buy the solution book
jyotirmayee
from where
Nirupam
where do u live ,,,,,if u live in Delhi then at bellsarayeiii or from stationary store ,,,,chatarpurrr ,,,,,there is a popular books store,,,,,u have to buy from there
jyotirmayee
Kota Rajasthan
Nirupam
so strange,,,,,r u preparing for pmt,,,?
jyotirmayee
for IIT
Nirupam
then concern near book store
jyotirmayee
in Rajasthan
jyotirmayee
or ask questions here
jyotirmayee
thanks for your help
Nirupam
its fine,,most welcome
jyotirmayee
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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