Question

How do you find instantaneous velocity in calculus?

Answer 1

Hint : The instantaneous velocity, also known as simply velocity, is a quantity that tells us how fast an object is moving somewhere along its course. It's the average velocity between two points on a path in the limit where the time (and thus the distance) between them reaches zero.

Complete step by step solution:
We can find the instantaneous velocity in calculus using the instantaneous velocity formula.
The formula is as follows:
$$ ${V_{\operatorname{int} }}\, = \,{\lim _{\vartriangle t \to 0}}\,\dfrac{{\Delta x}}{{\Delta t}}\, = \dfrac{{dx}}{{dt}}$
Where:
${V_{\operatorname{int} }}\,$ is the instantaneous velocity
$\Delta t$ is the time interval
$x$ is the variable of displacement
$t$ is the time
Example:
What is the Instantaneous Velocity of a particle moving in a straight line for $3$ seconds with a position function x of $5{t^2} + 2t + 4$ ?
Solution;
We have the function $x = 5{t^{^2}} + 2t + 4$ ----(1)
We know that the formula to compute instantaneous velocity is ${V_{\operatorname{int} }} = \dfrac{{dx}}{{dt}}$ -----(2)
Substituting (1) in (2) we get,
${V_{\operatorname{int} }} = \,\dfrac{d}{{dt}}(5{t^2} + 2t + 4)$
Differentiating we get,
${V_{\operatorname{int} }} = \,10t + 2$
Substituting the value of t as t=3, we get the instantaneous velocity as,
${V_{\operatorname{int} \,}} = 10 \times 3 + 2$
${V_{\operatorname{int} }} = 32m{s^{ - 1}}$
Therefore, we may conclude that the instantaneous velocity of the given function is $32m{s^{ - 1}}$

Note : The instantaneous velocity is the rate at which a single point's location or displacement $(x,t)$ changes with respect to time. whereas, Average velocity is the average rate of change of position (or displacement) with respect to time over a period of time.