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(C) Average velocity $V_1$ is defined as the total displacement divided by the total time interval.Given $x(t) = \alpha t^3 + \beta t^2 + \gamma$.At $

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$\frac{13 \alpha + 5 \beta}{27 \alpha + 6 \beta}$

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(C) Average velocity $V_1$ is defined as the total displacement divided by the total time interval.Given $x(t) = \alpha t^3 + \beta t^2 + \gamma$.At $t = 1 \ s$, $x(1) = \alpha(1)^3 + \beta(1)^2 + \gamma = \alpha + \beta + \gamma$.At $t = 3 \ s$, $x(3) = \alpha(3)^3 + \beta(3)^2 + \gamma = 27\alpha + 9\beta + \gamma$.Displacement $\Delta x = x(3) - x(1) = (27\alpha + 9\beta + \gamma) - (\alpha + \beta + \gamma) = 26\alpha + 8\beta$.Time interval $\Delta t = 3 - 1 = 2 \ s$.$V_1 = \frac{\Delta x}{\Delta t} = \frac{26\alpha + 8\beta}{2} = 13\alpha + 4\beta$.Instantaneous velocity $V_2$ is the derivative of position with respect to time: $V(t) = \frac{dx}{dt} = 3\alpha t^2 + 2\beta t$.At $t = 3 \ s$, $V_2 = 3\alpha(3)^2 + 2\beta(3) = 27\alpha + 6\beta$.The ratio $\frac{V_1}{V_2} = \frac{13\alpha + 4\beta}{27\alpha + 6\beta}$.

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