Question

You are participating in a road rally with your local classic car club. As you answer the questions that follow, remember that speed is how fast an object moves and velocity is the rate at which an object changes position.

1. You drive 30 miles due east in a half hour. Then you turn left and drive 30 miles north in
1 hour. What are your average speed and velocity, and what are the rectangular and polar coordinates of your position?

Answer 1

Average Speed: 60 miles/hour

Average Velocity: 34.6 miles/hour at a direction of 30 degrees North of East

Rectangular Coordinates: (0, 30) miles

Polar Coordinates: (30 miles, 60 degrees)

Explanation:

Average speed is calculated by dividing the total distance traveled by the total time taken, which is (30 + 30) miles in 1.5 hours or 60 miles/hour.

Average velocity is calculated by dividing the total displacement by the total time taken. The displacement is the straight-line distance from the starting point to the ending point, which is the hypotenuse of a right triangle with legs of 30 miles and 30 miles. Using the Pythagorean theorem, the displacement is 42.4 miles. The direction of the displacement is the arctangent of the opposite leg over the adjacent leg, which is 45 degrees North of East. Since the displacement took 1.5 hours, the average velocity is 42.4 miles/1.5 hours or 28.3 miles/hour.

The rectangular coordinates of the final position are (0, 30) miles since the car moved 30 miles north from its starting point, but no distance east or west.

The polar coordinates of the final position can be found by using the displacement as the radius and the direction of the displacement as the angle. The radius is 42.4 miles and the angle is 45 degrees + 30 degrees = 75 degrees (since the car turned left). However, polar coordinates are usually given with respect to the positive x-axis, which is East. Therefore, the angle needs to be measured counterclockwise from East, which is 15 degrees. Thus, the polar coordinates are (42.4 miles, 15 degrees).