Question

What is the derivative definition of instantaneous velocity?

Answer 1

Hint: This type of problem depends on the concept of instantaneous velocity. Instantaneous velocity is the velocity of an object at exactly specified instant when it is travelling. Hence the time period which we consider here is very small. Instantaneous velocity can be defined as the rate of change of position for a very short time interval.

Complete step by step answer:
We know that Instantaneous velocity can be defined as the rate of change of position for a very short time interval. Hence, we can write the derivative definition of instantaneous velocity as:
\[\Rightarrow \text{Instantaneous velocity }\left( v \right)=\displaystyle \lim_{\Delta t \to 0}\dfrac{\Delta x}{\Delta t}=\dfrac{dx}{dt}\]
Here, we can see that the instantaneous velocity depends on time that means for every t there is a different velocity at that given instant t. Hence, instantaneous velocity is a variable and so we can consider it as a function of time.
For example, let us consider, a position function
\[\Rightarrow x=4{{t}^{3}}+2{{t}^{2}}+5t+20\]
Since, \[\text{Instantaneous velocity }\left( v \right)=\dfrac{dx}{dt}\],
\[\Rightarrow v=\dfrac{d}{dt}\left( 4{{t}^{3}}+2{{t}^{2}}+5t+20 \right)\]
\[\begin{align}
  & \Rightarrow v=4\dfrac{d}{dt}{{t}^{3}}+2\dfrac{d}{dt}{{t}^{2}}+5\dfrac{d}{dt}t+20 \\
 & \Rightarrow v=4\left( 3{{t}^{2}} \right)+2\left( 2t \right)+5\left( 1 \right) \\
 & \Rightarrow v=12{{t}^{2}}+4t+5 \\
\end{align}\]
So that we can write that the instantaneous velocity for the position function \[x=4{{t}^{3}}+2{{t}^{2}}+5t+20\] is \[v=12{{t}^{2}}+4t+5\].
Let us say we want to know the value of instantaneous velocity at \[t=10\text{ seconds}\] and the position is measured in meters (m). Hence, the unit for instantaneous velocity is \[\text{m/sec}\].
Hence, at \[t=10\text{ seconds}\]
\[\Rightarrow v=12\left( {{10}^{2}} \right)+4\left( 10 \right)+5=1200+40+5=1245\text{m/sec}\].

Note: In this type of question students may make mistakes at defining derivatives of instantaneous velocity. As instantaneous velocity is a function of time the derivative definition must be \[\dfrac{dx}{dt}\] as x is used to represent position function and t is used for instant that is time. Also students have to take care during defining the unit of instantaneous velocity.