# Relative Velocity in 2 Dimensions

## Purpose:

• To discover how velocities combine in 2 dimensions.

## Discussion:

Consider Think and Solve Problem #1a on p. 42 of the text:

"A boat is rowed at 8 km/h directly across a river that flows at 6 km/h... What is the resultant speed of the boat?"

(The problem of combining velocities in two dimension is also discussed in section 3.2 Velocity Vectors, on p. 30 of the text. In particular, see Fig. 3.3)

The phrase "boat is rowed at 8 km/h" means that the speedometer of the boat measures the boat's speed relative to the water, so that if there is no current the boat's speed relative to the riverbank will be 8 km/h. Of course, the speed of the water (current) is measured relative to the riverbank.

The text says that the answer to the above question is 10 km/h at an angle of 37o from the original direction of the boat. The answer is arrived at as shown at right. This means that:

velocity of boat relative to the riverbank = velocity of boat relative to the water + velocity of water relative to the riverbank

where we need to remember that the "+" sign above refers to vector addition.

The question is, does all of this really work? In this lab, you can simulate this situation using a motorized cart to represent the boat and a large piece of paper to represent the water. The lab table can represent the riverbank.

## Equipment:

 variable-speed motorized cart meter stick or metric tape stop watch 2 meters of bulletin-board paper

## Procedure:

Design an experiment that will test the text's claim concerning the way velocities add in two dimensions. Some suggested equipment is listed above - you don't have to use all of it, and if you will need additional equipment contact your instructor. Be sure to discuss your plans with your instructor before you begin.

## Questions:

1. A boat is aimed directly across a river and its speedometer says 10 km/h. the captain of the boat knows that the current has a velocity of 4 km/h. What is the speed of the boat relative to the river bank? In what direction is the boat moving (relative to the bank)? What would be the boat's speed relative to the river bank if the current has a velocity of 10 km/h?
2. An airplane's speedometer indicates that it is moving with a velocity of 120 m/s relative to the air. The compass indicates that the airplane is heading east. If the weather report says that there is a wind blowing toward the north at 30 m/s (relative to the Earth) at the plane's altitude, what is the airplane's velocity relative to the Earth? What would the plane's velocity be if the wind were blowing at 90 km/h toward the south?
3. Suzie is riding in a car moving at 20 m/s (relative to the Earth) when she throws a ball out the window (perpendicular to the car's direction of travel). If she can throw a ball at 30 m/s, how fast will the ball be moving (relative to the Earth) when Harry catches it? (Ignore air resistance effects.)

last update October 12, 2001 by JL Stanbrough