$\"\"$

Average velocity :

The average velocity is defined as the ratio of change in distance by change in time.

$\"v_{avg}=\\frac{\\Delta$ , where $\"\\Delta$ and $\"\\Delta$ .

Use the table of values to find the average velocity.

(a)

(i)

Find the average velocity over the interval $\"[1,$ .

$\"v_{avg}=\\frac{\\Delta$

$\"v_{avg}=\\frac{s(t_2)-y(t_1)}{t_2-t_1}\"$

Substitute $\"t_1=1\"$ and $\"t_2=3\"$ in the above expression.

$\"\\\\v_{avg}=\\frac{s(3)-s(1)}{3-1}\\\\\"$

Substitute $\"s(1)=1.4\"$ and $\"s(3)=10.7\"$ in the above expression.

$\"\\\\=\\frac{10.7-1.4}{2}\\\\$

$\"\\\\=4.65\"$

The average velocity over the time interval $\"[1,$ is $\"4.65\"$ m/s.

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(ii)

Find the average velocity over the interval $\"[2,$ .

$\"v_{avg}=\\frac{\\Delta$

$\"v_{avg}=\\frac{s(t_2)-y(t_1)}{t_2-t_1}\"$

Substitute $\"t_1=2\"$ and $\"t_2=3\"$ in the above expression.

$\"\\\\v_{avg}=\\frac{s(3)-s(2)}{3-2}\\\\\"$

Substitute $\"s(2)=5.1\"$ and $\"s(3)=10.7\"$ in the above expression.

$\"\\\\=\\frac{10.7-5.1}{1}\\\\$

$\"\\\\=5.6\"$

The average velocity over the time interval $\"[2,$ is $\"5.6\"$ m/s.

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(iii)

Find the average velocity over the interval $\"[3,$ .

$\"v_{avg}=\\frac{\\Delta$

$\"v_{avg}=\\frac{s(t_2)-y(t_1)}{t_2-t_1}\"$

Substitute $\"t_1=3\"$ and $\"t_2=5\"$ in the above expression.

$\"\\\\v_{avg}=\\frac{s(5)-s(3)}{5-3}\\\\\"$

Substitute $\"s(3)=10.7\"$ and $\"s(5)=25.8\"$ in the above expression.

$\"\\\\=\\frac{25.8-10.7}{2}\\\\$

$\"\\\\=7.55\"$

The average velocity over the time interval $\"[3,$ is $\"7.55\"$ m/s.

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(iv)

Find the average velocity over the interval $\"[3,$ .

$\"v_{avg}=\\frac{\\Delta$

$\"v_{avg}=\\frac{s(t_2)-y(t_1)}{t_2-t_1}\"$

Substitute $\"t_1=3\"$ and $\"t_2=4\"$ in the above expression.

$\"\\\\v_{avg}=\\frac{s(4)-s(3)}{4-3}\\\\\"$

Substitute $\"s(3)=10.7\"$ and $\"s(4)=17.7\"$ in the above expression.

$\"\\\\=\\frac{17.7-10.7}{1}\\\\$

$\"\\\\=7\"$

The average velocity over the time interval $\"[3,$ is $\"7\"$ m/s.

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(b)

Graph :

Graph of the function $\"s(t)\"$ by using table of values.

(1) Draw the coordinate plane.

(2) Plot the points from the table.

(3) Connect the plotted points to a smooth curve.

Draw the approximate tangent line at $\"t=3\"$ .

$\"\"$

From the graph, $\"\\Delta$ and $\"\\Delta$ .

Instantaneous velocity :

The instantaneous velocity is defined as $\"v=\\lim_{\\Delta$ .

$\"\\\\v=\\frac{12}{1.9}\\\\$

The instantaneous velocity when $\"t=3\"$ is $\"6.3\"$ m/s.

$\"\"$

(a) (i) The average velocity over the time interval $\"[1,$ is $\"4.65\"$ m/s.

(ii) The average velocity over the time interval $\"[2,$ is $\"5.6\"$ m/s.

(iii) The average velocity over the time interval $\"[3,$ is $\"7.55\"$ m/s.

(iv) The average velocity over the time interval $\"[3,$ is $\"7\"$ m/s.

(b) The instantaneous velocity when $\"t=3\"$ is $\"6.3\"$ m/s.