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# Velocity

## The speed of an object in a given direction.

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Practice Velocity
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Velocity

Students will learn how analyze and solve problems in two dimensions containing velocity vectors but no acceleration.

### Key Equations

\begin{align*} v_{x} = v \cos \theta\end{align*}, where \begin{align*}\theta\end{align*} is the angle between the velocity vector and the horizontal.

\begin{align*} v_{y} = v \sin \theta\end{align*}, where \begin{align*}\theta\end{align*} is the angle between the velocity vector and the horizontal.

\begin{align*}\Delta x = v_{x} t \end{align*}

\begin{align*}\Delta y = v_{y} t \end{align*}

Guidance
The only equation you need is that displacement in a certain direction equals the component of velocity in that direction multiplied by the time it takes. You'll use this once for the x-direction and once for the y-direction and solve for what is asked.

#### Example 1

Question: If a river is flowing north at 2 m/s and you swim straight across (i.e. east) at 1.5 m/s, how far up shore will you be from your starting point once you reach the other side? The river is 9 m wide.

Answer: First solve for the time it takes you to reach the other side. Let's let north be the y-direction and the direction across the river be the x-direction.

\begin{align*}\Delta x = v_{x} t \end{align*}

\begin{align*} 9 m = 1.5 m/s \times t \end{align*}

thus, \begin{align*} t = 6 s \end{align*}

Now, use the time you are in the water to find how far the river has carried you north.

\begin{align*}\Delta y = v_{y} t \end{align*}

\begin{align*}\Delta y = 2 m/s \times 6 s \end{align*}

\begin{align*}\Delta y = 12 m \end{align*}

### Time for Practice

1. If a river is flowing south at 4 m/s and you swim straight across (i.e. east) at 2 m/s; admittedly, you're going to drift a bit south. That said, calculate that distance that you drifted south from your starting point. The river is 16 m wide.
2. If a river is flowing south at 3 m/s and you swim at an angle of 30 degrees north of directly east at 1 m/s, how far did you drift up or down stream from your starting point once you reach the other side? The river is 10 m wide.
3. If a river is flowing north at 2 m/s and you can swim at 4 m/s, what angle should you swim at such that you arrive directly across the river (i.e. no drift north or south from starting point on other side)? The river is 10 m wide.
4. If a river is flowing south at 5 m/s and you can swim at 4 m/s maximum, is it possible to arrive directly across? Why or why not?

#### Answers to Selected Problems

1. \begin{align*}32 \;\mathrm{m}\end{align*}
2. \begin{align*}28.9 \;\mathrm{m}\end{align*} south of starting point
3. \begin{align*}30\end{align*} degrees
4. No, even if you swim directly north the river will still take you south at 1 m/s

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