$\"\"$

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The velocity of the ball is thrown into the air is $\"40\"$ ft/s.

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The function of the ball is $\"y=40t-16t^2\"$ .

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(a)

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Average velocity :

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

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

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Now ball is at beginning stage at $\"t=2\"$ .

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Substitute $\"t=2\"$ in the function $\"y(t)=40t-16t^2\"$ .

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$\"\\\\y(2)=40(2)-16(2)^2\\\\$

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$\"\"$

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(i)

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Find the average velocity at $\"\\Delta$ .

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

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$\"v_{avg}=\\frac{y(t+\\Delta$

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Substitute $\"t=2\"$ and $\"\\Delta$ in the above expression.

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$\"\\\\v_{avg}=\\frac{y(2+0.5)-y(2)}{0.5}\\\\\"$

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$\"\\\\=\\frac{y(2.5)-y(2)}{0.5}\\\\$

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$\"\\\\=-32\"$

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The average velocity at $\"\\Delta$ is $\"-32\"$ ft/s.

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$\"\"$

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(ii)

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Find the average velocity at $\"\\Delta$ .

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

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$\"v_{avg}=\\frac{y(t+\\Delta$

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Substitute $\"t=2\"$ and $\"\\Delta$ in the above expression.

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$\"\\\\v_{avg}=\\frac{y(2+0.1)-y(2)}{0.1}\\\\\"$

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$\"\\\\=\\frac{y(2.1)-y(2)}{0.1}\\\\$

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$\"\\\\=-25.6\"$

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The average velocity at $\"\\Delta$ is $\"-25.6\"$ ft/s.

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$\"\"$

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(iii)

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Find the average velocity at $\"\\Delta$ .

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

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$\"v_{avg}=\\frac{y(t+\\Delta$

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Substitute $\"t=2\"$ and $\"\\Delta$ in the above expression.

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$\"\\\\v_{avg}=\\frac{y(2+0.05)-y(2)}{0.05}\\\\\"$

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$\"\\\\=\\frac{y(2.05)-y(2)}{0.05}\\\\$

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$\"\\\\=-24.8\"$

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The average velocity at $\"\\Delta$ is $\"-24.8\"$ ft/s.

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$\"\"$

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(iv)

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Find the average velocity at $\"\\Delta$ .

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

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$\"v_{avg}=\\frac{y(t+\\Delta$

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Substitute $\"t=2\"$ and $\"\\Delta$ in the above expression.

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$\"\\\\v_{avg}=\\frac{y(2+0.01)-y(2)}{0.01}\\\\\"$

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$\"\\\\=\\frac{y(2.01)-y(2)}{0.01}\\\\$

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$\"\\\\=-24.16\"$

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The average velocity at $\"\\Delta$ is $\"-24.16\"$ ft/s.

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$\"\"$

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(b)

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Find the instantaneous velocity when $\"t=2\"$ .

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Instantaneous velocity :

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

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

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Substitute $\"t=2\"$ in the above expression.

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

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$\"=\\lim_{\\Delta$ ( Since $\"y(t)=40t-16t^2\"$ )

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

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

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

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

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

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$\"\\\\={-24+16(0)}\\\\$

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The instantaneous velocity when $\"t=2\"$ is $\"-24\"$ ft/s.

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$\"\"$

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(a) (i) The average velocity at $\"\\Delta$ is $\"-32\"$ ft/s.

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(ii) The average velocity at $\"\\Delta$ is $\"-25.6\"$ ft/s.

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(iii) The average velocity at $\"\\Delta$ is $\"-24.8\"$ ft/s.

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(iv) The average velocity at $\"\\Delta$ is $\"-24.16\"$ ft/s.

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(b) The instantaneous velocity when $\"t=2\"$ is $\"-24\"$ ft/s.