Some rental cars have a GPS unit installed, which allows the...

Some rental cars have a GPS unit installed, which allows the rental car company to check where you are at all times and thus also know your speed at any time. One of these rental cars is driven by an employee in the company's lot and, during the time interval from 0 to $10 \mathrm{~s}$, is found to have a position vector as a function of time of $$ \begin{aligned} \vec{r}(t)=&\left((24.4 \mathrm{~m})-t(12.3 \mathrm{~m} / \mathrm{s})+t^{2}\left(2.43 \mathrm{~m} / \mathrm{s}^{2}\right)\right.\\ &\left.(74.4 \mathrm{~m})+t^{2}\left(1.80 \mathrm{~m} / \mathrm{s}^{2}\right)-t^{3}\left(0.130 \mathrm{~m} / \mathrm{s}^{3}\right)\right) \end{aligned} $$ a) What is the distance of this car from the origin of the coordinate system at $t=5.00 \mathrm{~s} ?$ b) What is the velocity vector as a function of time? c) What is the speed at $t=5.00 \mathrm{~s} ?$ Extra credit: Can you produce a plot of the trajectory of the car in the $x y$ -plane?

Some rental cars have a GPS unit installed, which allows the rental car company to check where you are at all times and thus also know your speed at any time. One of these rental cars is driven by an employee in the company's lot and, during the time interval from 0 to $10 \mathrm{~s}$, is found to have a position vector as a function of time of
$$
\begin{aligned}
\vec{r}(t)=&\left((24.4 \mathrm{~m})-t(12.3 \mathrm{~m} / \mathrm{s})+t^{2}\left(2.43 \mathrm{~m} / \mathrm{s}^{2}\right)\right.\\
&\left.(74.4 \mathrm{~m})+t^{2}\left(1.80 \mathrm{~m} / \mathrm{s}^{2}\right)-t^{3}\left(0.130 \mathrm{~m} / \mathrm{s}^{3}\right)\right)
\end{aligned}
$$
a) What is the distance of this car from the origin of the coordinate system at $t=5.00 \mathrm{~s} ?$
b) What is the velocity vector as a function of time?
c) What is the speed at $t=5.00 \mathrm{~s} ?$
Extra credit: Can you produce a plot of the trajectory of the car in the $x y$ -plane?

Key Concepts

Euclidean Distance Calculation

The Euclidean distance from the origin to a point in the plane is determined using the formula derived from the Pythagorean theorem, where the distance is the square root of the sum of the squares of the x and y coordinates. This concept is essential in determining how far the car is from the origin at a given time.

Parametric Equations for Trajectory Plotting

Parametric equations express the coordinates of points on a curve as functions of a variable, such as time. This allows the trajectory of a moving object to be plotted in the coordinate plane by eliminating the parameter. Understanding parametric representation is vital for visualizing the path of motion, as it provides a clear picture of the object's trajectory over time.

Differentiation to Obtain Velocity

Differentiation is the process used to derive the velocity vector from the position vector. By taking the derivative of each component of the position vector with respect to time, the velocity vector, which indicates both the speed and direction of motion, is obtained. This process is fundamental in calculus-based physics to transition from position to velocity.

Position Vector

A position vector is a function that represents the location of an object in space as a function of time. In problems like this, the position vector provides the (x, y) coordinates of the moving object at any given time, and it serves as the basis for deriving other kinematic quantities such as displacement, velocity, and acceleration.

Speed as the Magnitude of Velocity

Speed is the scalar measure of how fast an object is moving, irrespective of its direction. It is calculated as the magnitude of the velocity vector. In this context, computing the speed involves taking the square root of the sum of the squares of the velocity components, which is a common application of the Pythagorean theorem in kinematics.

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