Question

@bobross plz help I have another question
A particle moves on a line away from its initial position so that after t hours it is s = 6t^2 + 2t miles from its initial position. Find the average velocity of the particle over the interval [1, 4]. Include units in your answer.

Answer 1

Answer:

Average velocity is 32 miles/hr.

Step-by-step explanation:

Given that a particle moves on a line away from its initial position so that after t hours it is  $s=6t^2+2t$ miles from its initial position.

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

As, average velocity is the change is position over the change in time.

$s(4)=6(4)^2+2(4)=104$

$s(1)=6(1)^2+2(1)=8$

∴ $\text{Average Velocity=}\frac{s(4)-s(1)}{4-1}$

                            =$\frac{104-8}{3}=\frac{96}{3}=32miles/hr$

Hence, average velocity is 32 miles/hr.

                                         

Answer 2

The average velocity of the 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 $s\left(t\right)=4{t^2}+t$ . The given interval is $\left[{1,4}\right]$ .

Step by step explanation:

Step 1:  

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

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

The distance travelled by the particle 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 particle at $t=4$  is as follows,

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

Now find the distance travelled by the particle in the 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 given interval is $\left[{1,4}\right]$ .

Now we need to find the time as,

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

Step 3:  

Now we find the average velocity of the particle.

The average velocity can be calculated as,

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

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

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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.