We define the variables:
• s = position of a particle in ml,
,• t = time in h.
From the statement of the problem, we know that:
• a particle moves in a straight line,
,• its position from the initial position is given by the function:
[tex]s(t)=4t^2+t.[/tex]We must find the average velocity of the particle over the time interval [1,4].
The average velocity is defined as the change in position or displacement (∆s) divided by the time interval (∆t) in which the displacement occurs.
The time interval is:
[tex]\Delta t=t_2-t_1=4-1=3.[/tex]The displacement is:
[tex]\Delta s=s_2-s_1=s(t_2)-s(t_1)=s(4)-s(1)=(4\cdot4^2+4)-(4\cdot1^2+1)=68-5=63.[/tex]Using the definition of the average velocity, we get:
[tex]\bar{v}=\frac{\Delta s}{\Delta t}=\frac{63}{3}=21.[/tex]Now, the units of the velocity are distance (in ml) over time (in h), so we have:
[tex]\bar{v}=21\cdot\frac{ml}{h}.[/tex]Answer
The average velocity of the particle in the time interval [1,4] is:
[tex]\bar{v}=21\cdot\frac{ml}{h}.[/tex]