$\"\"$

The equation of displacement of a particle moving back and fourth along a straight line is $\"s=2\\sin$ .

(a)

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 $\"\\Delta$ .

(i)

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

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

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

Substitute $\"t_1=1\"$ and $\"t_2=2\"$ in the above expression.

$\"\\\\v_{avg}=\\frac{s(2)-s(1)}{2-1}\\\\\"$

$\"\\\\=\\frac{[2\\sin$

$\"=6\"$

The average velocity over the time interval $\"[1,$ is $\"6\"$ m/s.

$\"\"$

(ii)

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

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

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

Substitute $\"t_1=1\"$ and $\"t_2=1.1\"$ in the above expression.

$\"\\\\v_{avg}=\\frac{s(1.1)-s(1)}{1.1-1}\\\\\"$

$\"\\\\=\\frac{[2\\sin$

$\"=-4.711\"$

The average velocity over the time interval $\"[1,$ is $\"-4.711\"$ m/s.

$\"\"$

(iii)

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

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

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

Substitute $\"t_1=1\"$ and $\"t_2=1.01\"$ in the above expression.

$\"\\\\v_{avg}=\\frac{s(1.01)-s(1)}{1.01-1}\\\\\"$

$\"\\\\=\\frac{[2\\sin$

$\"=-6.13\"$

The average velocity over the time interval $\"[1,$ is $\"-6.13\"$ m/s.

$\"\"$

(iv)

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

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

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

Substitute $\"t_1=1\"$ and $\"t_2=1.001\"$ in the above expression.

$\"\\\\v_{avg}=\\frac{s(1.001)-s(1)}{1.001-1}\\\\\"$

$\"\\\\=\\frac{[2\\sin$

$\"=-6.2\"$

The average velocity over the time interval $\"[1,$ is $\"-6.2\"$ m/s.

$\"\"$

(b)

Find the instantaneous velocity when $\"t=1\"$ .

Instantaneous velocity :

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

$\"v=\\lim_{\\Delta$

Substitute $\"t=1\"$ in the above expression.

$\"v=\\lim_{\\Delta$

$\"=\\lim_{\\Delta$ ( Since $\"s(t)=2\\sin$ )

$\"=\\lim_{\\Delta$

$\"=\\lim_{\\Delta$

$\"=\\lim_{\\Delta$

$\"=\\lim_{\\Delta$

$\"=\\lim_{\\Delta$

$\"=\\lim_{\\Delta$

$\"=\\lim_{\\Delta$

$\"=\\left$

$\"\"$

(a) (i) The average velocity over the time interval $\"[1,$ is $\"6\"$ m/s.

(ii) The average velocity over the time interval $\"[1,$ is $\"-4.711\"$ m/s.

(iii) The average velocity over the time interval $\"[1,$ is $\"-6.13\"$ m/s.

(iv) The average velocity over the time interval $\"[1,$ is $\"-6.2\"$ m/s.

(b)