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.