Question

How do you find the instantaneous velocity at t=2 for the position function $s\left( t \right) = {t^3} + 8{t^2} - t$ ?

Answer 1

Instantaneous velocity is defined as the rate of change of position for a time interval which is very small (almost zero). Measured using SI unit m/s. We can find instantaneous velocity by finding its derivative with respect to t, as the position function is given hence by finding $\dfrac{{ds}}{{dt}}$ we can get the velocity.

Complete step by step answer:
The given function is
$s\left( t \right) = {t^3} + 8{t^2} - t$
The instantaneous velocity is given by $\dfrac{{ds}}{{dt}}$, hence find the derivative of the given function with respect to t as
$\dfrac{{ds}}{{dt}} = {t^3} + 8{t^2} - t$
We get the derivative as
$\dfrac{{ds}}{{dt}} = 3{t^2} + 8\left( {2t} \right) - 1$ ……………….. 1
As the value of t is given as t = 2, hence substituting the value in equation 1 we get
${\left[ {\dfrac{{ds}}{{dt}}} \right]_{t = 2}} = 3{t^2} + 8\left( {2t} \right) - 1$
$\Rightarrow{\left[ {\dfrac{{ds}}{{dt}}} \right]_{t = 2}} = 3{\left( 2 \right)^2} + 8\left( {2\left( 2 \right)} \right) - 1$
Simplifying the terms, we get
${\left[ {\dfrac{{ds}}{{dt}}} \right]_{t = 2}} = 3\left( 4 \right) + 8\left( 4 \right) - 1$
$\Rightarrow{\left[ {\dfrac{{ds}}{{dt}}} \right]_{t = 2}} = 12 + 32 - 1$
$\therefore{\left[ {\dfrac{{ds}}{{dt}}} \right]_{t = 2}} = 43$

Therefore, the instantaneous velocity at t=2 is 43.

Additional information:
Instantaneous speed is the magnitude of the instantaneous velocity. It has the same value as that of instantaneous velocity but does not have any direction. The velocity of an object at that instant of time. Instantaneous velocity definition is given as “The velocity of an object under motion at a specific point of time.”

It is determined very similarly as that of average velocity, but here the time period is narrowed. We know that the average velocity for a given time interval is total displacement divided by total time. As this time interval approaches zero, the displacement also approaches zero. But the limit of the ratio of displacement to time is non-zero and is called instantaneous velocity.

Note: The key point to find the instantaneous velocity is to find the derivative of s(t), as the instantaneous velocity is given by $\dfrac{{ds}}{{dt}}$, and an average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero.