Distance from velocity data The accompanying table gives...

Distance from velocity data The accompanying table gives data for the velocity of a vintage sports car accelerating from 0 to $142 \mathrm{mi} / \mathrm{h}$ in $36 \mathrm{sec}(10$ thousandths of an hour $)$ $$ \begin{array}{lc|cc} \hline \begin{array}{c} \text { Time } \\ \text { (h) } \end{array} & \begin{array}{c} \text { Velocity } \\ \text { (mi/h) } \end{array} & \begin{array}{c} \text { Time } \\ \text { (h) } \end{array} & \begin{array}{c} \text { Velocity } \\ \text { (mi / h) } \end{array} \\ \hline 0.0 & 0 & 0.006 & 116 \\ 0.001 & 40 & 0.007 & 125 \\ 0.002 & 62 & 0.008 & 132 \\ 0.003 & 82 & 0.009 & 137 \\ 0.004 & 96 & 0.010 & 142 \\ 0.005 & 108 & & \\ \hline \end{array} $$ a. Use rectangles to estimate how far the car traveled during the 36 sec it took to reach $142 \mathrm{mi} / \mathrm{h}$ b. Roughly how many seconds did it take the car to reach the halfway point? About how fast was the car going then?

Distance from velocity data The accompanying table gives data for the velocity of a vintage sports car accelerating from 0 to $142 \mathrm{mi} / \mathrm{h}$ in $36 \mathrm{sec}(10$ thousandths of an hour $)$
$$
\begin{array}{lc|cc}
\hline \begin{array}{c}
\text { Time } \\
\text { (h) }
\end{array} & \begin{array}{c}
\text { Velocity } \\
\text { (mi/h) }
\end{array} & \begin{array}{c}
\text { Time } \\
\text { (h) }
\end{array} & \begin{array}{c}
\text { Velocity } \\
\text { (mi / h) }
\end{array} \\
\hline 0.0 & 0 & 0.006 & 116 \\
0.001 & 40 & 0.007 & 125 \\
0.002 & 62 & 0.008 & 132 \\
0.003 & 82 & 0.009 & 137 \\
0.004 & 96 & 0.010 & 142 \\
0.005 & 108 & & \\
\hline
\end{array}
$$
a. Use rectangles to estimate how far the car traveled during the 36 sec it took to reach $142 \mathrm{mi} / \mathrm{h}$
b. Roughly how many seconds did it take the car to reach the halfway point? About how fast was the car going then?

Key Concepts

Unit Conversion

Unit conversion involves changing quantities from one unit of measure to another, which is crucial in solving real-world problems where data may be provided in different units. Correct conversion ensures that calculations such as integrations or derivations are performed consistently and accurately.

Velocity-Displacement Relationship

The displacement of an object can be determined by integrating its velocity over time, as displacement is the area under the velocity versus time curve. This relationship is foundational in kinematics, connecting the concept of rate (velocity) with a cumulative change (distance traveled).

Interpolation

Interpolation is the method of estimating values between known data points. In contexts where data is sampled at discrete intervals, interpolation allows one to approximate values at intermediate points, which can be especially useful for determining when a certain threshold is met in a process, such as reaching half a target value.

Numerical Integration

Numerical integration is a method used to approximate the definite integral of a function when an antiderivative is not easily obtainable or when only discrete data is available. This technique involves summing the contributions of small segments to estimate the area under a curve, which in many applications corresponds to a physical quantity like distance or work.

Riemann Sum

A Riemann sum is a specific numerical integration method where the area under a curve is approximated by summing the areas of rectangular slices. Each rectangle's height is determined by the value of the function at a chosen sample point within the subinterval, and this method provides a way to estimate integrals from discrete data points.

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