We are asked to find the *position*, *average velocity*, and *instantaneous velocity* for a ball, given its position function.

Remember that to find an object's position at a certain time, we just put that value of *t* into the function and calculate the result (as long as everything is in the right units).

To calculate the average velocity, we use the formula

$\overline{){\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}}_{\mathit{a}\mathit{v}\mathit{g}}{\mathbf{=}}\frac{\mathbf{\u2206}\stackrel{\mathbf{\rightharpoonup}}{\mathit{x}}}{\mathbf{\u2206}\mathit{t}}{\mathbf{=}}\frac{{\mathit{x}}_{\mathbf{2}}\mathbf{-}{\mathit{x}}_{\mathbf{1}}}{{\mathit{t}}_{\mathbf{2}}\mathbf{-}{\mathit{t}}_{\mathbf{1}}}}$

Anytime we're given a **position**, **velocity**, or **acceleration** function and asked to *find one or more of the others*, we know it's a **motion problem with calculus**. A PVA diagram like the one below can help remind you of the relationships between the three functions:

$\mathit{P}\begin{array}{c}{\mathbf{\leftarrow}}\\ {\mathbf{\to}}\end{array}\underset{\frac{\mathit{d}}{\mathit{d}\mathit{t}}}{\overset{{\mathbf{\int}}{\mathit{d}}{\mathit{t}}}{\mathit{V}}}\begin{array}{c}{\mathbf{\leftarrow}}\\ {\mathbf{\to}}\end{array}\mathit{A}$

To get from position to velocity, we take the derivative of the position function.

$\overline{){\mathit{v}}{\mathbf{\left(}}{\mathit{t}}{\mathbf{\right)}}{\mathbf{=}}\frac{\mathit{d}\mathit{x}\mathbf{\left(}\mathit{t}\mathbf{\right)}}{\mathit{d}\mathit{t}}}$

We'll also need to remember the __power rule of derivation__, which is:

$\overline{)\frac{\mathit{d}}{\mathit{d}\mathit{t}}\mathbf{\left(}{\mathit{t}}^{\mathit{n}}\mathbf{\right)}{\mathbf{=}}{\mathit{n}}{{\mathit{t}}}^{\mathit{n}\mathbf{-}\mathbf{1}}}$ (for example, $\frac{\mathit{d}}{\mathit{d}\mathit{t}}{\mathit{t}}^{\mathbf{3}}\mathbf{=}\mathbf{3}{\mathit{t}}^{\mathbf{2}}$)

The position of a ball rolling in a straight line is given by *x*(*t*) = 2.1 − 3.7*t* + 3.3*t*^{2}, where *x* is in meters and *t* in seconds. **(a)** Determine the position of the ball at *t* = 2.0 s. **(b)** What is the average velocity from *t* = 2.0 s to *t* = 5.0 s? **(c)** What is its instantaneous velocity at *t* = 5.0 s?

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