Position Functions and Instantaneous Velocity Video Lessons

Concept

# Problem: The position of a ball rolling in a straight line is given by x(t) = 2.1 − 3.7t + 3.3t2, 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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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

$\overline{){\stackrel{\mathbf{⇀}}{\mathbit{v}}}_{\mathbit{a}\mathbit{v}\mathbit{g}}{\mathbf{=}}\frac{\mathbf{∆}\stackrel{\mathbf{⇀}}{\mathbit{x}}}{\mathbf{∆}\mathbit{t}}{\mathbf{=}}\frac{{\mathbit{x}}_{\mathbf{2}}\mathbf{-}{\mathbit{x}}_{\mathbf{1}}}{{\mathbit{t}}_{\mathbf{2}}\mathbf{-}{\mathbit{t}}_{\mathbf{1}}}}$

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:

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

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

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

We'll also need to remember the power rule of derivation, which is:

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

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###### Problem Details

The position of a ball rolling in a straight line is given by x(t) = 2.1 − 3.7t + 3.3t2, 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?