Problems on Instantaneous Velocity
A speedometer shows different Speeds at different instants
This is because vehicles don't travel with constant Speed at all times
At any given instance what we read from the speedometer of the vehicle is instantaneous speed
Instantaneous speed can be defined as the speed of an object at a particular time or instant
And if we include the direction with the instantaneous Speed, we get the Instantaneous Velocity of the object
As per definition, Instantaneous Velocity is the rate of change of Displacement per unit Time
We can express it mathematically as,
$InstantaneousVelocity=ChangeinTimeChangeinDisplacement $
$∴v=dtdx $
$v=dtdx $
Where,
$v=Instantaneous Velocitydtdx =Rate of change of Displacement$
To understand the concept better, let us solve a problem using this equation
Problem
An object moving with a Velocity
$v=(3t_{2}+2)$
. Calculate the Displacement of the object between
$t=1s$
and
$t=2s$
Solution
As per definition, we know that Velocity is the rate of change of position of an object in unit Time
Therefore we can say,
$Velocity=TimeDisplacement $
Velocity
$v=dtdx $
We can also say that,
$dtdx =3t_{2}+2⇒dx=(3t_{2}+2)dt$
Integrating L.H.S with limits
$x_{1}$
to
$x_{2}$
and integrating R.H.S with limits Time
$t=1s$
to
$t=2s$
,
$∫_{x_{1}}dx=∫_{t=1}(3t_{2}+2)dt$
$⇒[x]_{x_{1}}=3∫_{1}t_{2}+2∫_{1}dt⇒x_{2}−x_{1}=3[3t_{3} ]_{1}+2[t]_{1}$
Change in displacement
$△x=[2_{3}−1]+2[2−1]⇒△x=(8−1)+2⇒△x=7+2∴△x=9m$
Therefore, the Displacement made by the object in time
$t=1s$
to
$t=2s$
is
$9m$
Revision
Instantaneous Velocity is the Velocity of an object in Motion at a specific point in Time
Instantaneous velocity can be mathematically expressed as,
$v=dtdx $
Where,
$dtdx =RateofchangeofVelocity$
The End