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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

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

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

$\Delta x = v_{x} t$

$\Delta y = v_{y} t$

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.

$\Delta x = v_{x} t$

$9 m = 1.5 m/s \times t$

thus, $t = 6 s$

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

$\Delta y = v_{y} t$

$\Delta y = 2 m/s \times 6 s$

$\Delta y = 12 m$

### 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?

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