## Statement

difficulty Knowing that the distance a body travels as a function of time is given by the following equation:

$S\left(t\right)={t}^{2}+5·t+1$

Calculate:

a) The average speed during the first 3 seconds.
b) The instantaneous speed of the body.

## Solution

Question a)

Data

The equation of distance traveled as a function of time:

$S\left(t\right)={t}^{2}+5·t+1$

The instants of time where to study the speed:

t1=0 sg y t2=3 sg

Resolution

To calculate the average speed, we must use the following equation:

We know the times (t1 and t2) between which we will calculate the speed, but we do not know directly the distance that the body has traveled in each of them. However, since we have the function that determines how much distance it has traveled at every moment of time, it will be enough to substitute both values of t in the function to find out.

For t1 = 0 s

For t2=3 sg

At this point, we have all the necessary information to calculate the average speed in the interval between 0 and 3 seconds, substituting in the first equation:

Question b)

Instantaneous speed is calculated by solving the following equation:

$V =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆s}{∆t}$

To calculate Δs, we will take the time t1 = t, and t2 = t + ΔtPara calcular Δs, vamos a tomar como tiempo t1=t y t2=t+Δt:

By applying the formula of instantaneous speed:

$V\left(t\right)=\underset{\Delta t\to 0}{\mathrm{lim}}\frac{\Delta {t}^{2}+2·t·\Delta t+5·\Delta t}{\Delta t}⇒\phantom{\rule{0ex}{0ex}}\overline{)V\left(t\right)=2·t+5}\phantom{\rule{0ex}{0ex}}$

## Formulas worksheet

These are the main formulas that you must know to solve this exercise. If you are not clear about their meaning, we recommend you to check the theory in the corresponding sections. Furthermore, you will find in them, under the Formulas tab, the codes that will allow you to integrate these equations in external programs like Word or Mathematica.

Formulas
Related sections
$V =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆s}{∆t}$