Question

A particle moves on a line away from its initial position so that after t hours it is s = 4t^2 + t miles from its initial position. Find the average velocity of the particle over the interval [1, 4]. Include units in your answer.

Answer 1

The average velocity of the moving particle is $\boxed{{\mathbf{21 units}}}$ .

Further explanation:  

Velocity is the speed of an object in a given direction. Velocity is the vector quantity.

The average velocity can be calculated as,

  ${\text{average velocity}}=\frac{{{\text{distance travelled}}}}{{{\text{time taken}}}}$

Given:

The position of the particle after $t$ hours is $s\left(t\right)=4{t^2}+t$ and the interval is $\left[{1,4}\right]$ .

Step by step explanation:

Step 1:  

The given position of the particle after $t$  hours is $s\left(t\right)=4{t^2}+t$ .

First find the distance travelled in the interval of $\left[{1,4}\right]$ .

The distance travelled by the moving particle in a line at $t=1$  is as follows,

$\begin{gathered}s\left(t\right)=4{\left(t\right)^2}+t\hfill\\s\left(1\right)=4{\left(1\right)^2}+1\hfill\\s\left(1\right)=5\hfill\\\end{gathered}$

The distance travelled by the moving particle in a line at $t=4$  is as follows,

$\begin{gathered}s\left(t\right)=4{\left(t\right)^2}+t\hfill\\s\left(4\right)=4{\left(4\right)^2}+4\hfill\\s\left(1\right)=68\hfill\\\end{gathered}$

Now evaluate the total distance travelled by the moving particle in the given interval of $\left[{1,4}\right]$ .

$\begin{aligned}{\text{distance travelled}}&=s\left(4\right)-s\left(1\right)\\&=68-5\\&=63\\\end{aligned}$

Step 2:  

The provided interval is $\left[{1,4}\right]$ .

Now the time can be calculated as,

$\begin{aligned}{\text{time elapsed}}&=4-1\\&=3\\\end{aligned}$

Step 3:  

Now we evaluate the average velocity of the moving particle.

The average velocity can be evaluated as,

$\begin{aligned}{\text{average velocity}}&=\frac{{{\text{distance travelled}}}}{{{\text{time elapsed}}}}\\&=\frac{{63}}{3}\\&=21{\text{units}}\\\end{aligned}$

Thus, the average velocity of the moving particle is $21{\text{ units}}$ .

Learn more:  

• Learn more about the distance between the points brainly.com/question/6278187

• Learn more about the equivalent fraction brainly.com/question/952259

• Learn more about midpoint of the segment brainly.com/question/3269852

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Speed, distance and time

Keywords: velocity, initial position, particle, moves, interval, distance travelled, time elapsed, position, average velocity, units, vector quantity, speed, direction, hours.

Answer 2

Answer:

Average velocity is 21 miles/hr

Step-by-step explanation:

Given a particle moving away from its initial position

Position after t hours that is

                    s(t) = 4$(t^{2})$ + t

We have to find the average velocity of the particle over the interval [1,4]

Average velocity = $\frac{Distance Travelled}{Time Elapsed}$

             Distance travelled = $[s]_{1}^{4}$

                                             = s(4) - s(1)

                                             = 4$(4^{2})$ + 4 - 4$(1^{2})$ -1

                                             =  63 miles

                Time elapsed = 4 - 1 = 3

Average velocity = $\frac{63}{3}$

                             = 21 miles/hr