Question

How to calculate the Instantaneous speed?

Answer 1

Instantaneous speed is the speed of a particle or object at that particular instant of time, like whenever you check the speedometer of your vehicle it shows you the Instantaneous speed (speed for that instant of time). Formula for instantaneous speed is
${{S}_{inst.}}=\underset{t\to T}{\mathop{\lim }}\,\dfrac{dx}{dt}$

To find Instantaneous Speed we need to differentiate a particular given equation with respect to time, because we need the speed for that particular time, hence we are differentiating it with respect to time.

Complete Solution:
To find the instantaneous speed you need to have a equation of motion or some displacement function

For example, if we are asked to find the Instantaneous speed for a given function $x(t)=5{{t}^{2}}+10t+6$ at T=5 seconds.

Then we will use the formula
${{S}_{inst.}}=\underset{t\to T}{\mathop{\lim }}\,\dfrac{dx}{dt}$
Where t is the time variable of equation and,
T is the time at which we need to find out the instantaneous speed and
X is the equation given in the question
${{S}_{inst.}}$is the required instantaneous speed at that instant of time.

Now we need to put the value of the equation in our formula
${{S}_{inst.}}=\underset{t\to 5}{\mathop{\lim }}\,\dfrac{d(5{{t}^{2}}+10t+6)}{dt}$

We need to differentiate the above equation or we can say that we need to solve the above equation,after differentiation we get,
${{S}_{inst.}}=\underset{t\to 5}{\mathop{\lim }}\,10t+10$

Now Put the limit and we get,
${{S}_{inst.}}=10(5)+10$
We get ${{S}_{inst.}}=60m{{s}^{-1}}$

Hence, The Instantaneous Speed for the above question is ${{S}_{inst.}}=60m{{s}^{-1}}$.

Note:
Do not get confused between Average and instantaneous speed, they both are different. Average speed tells you the mean speed for the given interval of time, while instantaneous speed tells you the speed for that particular instant of time.
Whenever you see the speed in a speedometer, it is for that particular instant of time hence that is instantaneous speed not average speed.