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

### Key Equations

, where is the angle between the velocity vector and the horizontal.

, where is the angle between the velocity vector and the horizontal.

#### 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.

thus,

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

### Watch this Explanation

### Simulation

### Explore More

- 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.
- 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.
- 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.
- 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

- south of starting point
- degrees
- No, even if you swim directly north the river will still take you south at 1 m/s