Chapter 3: Two-Dimensinal Kinematics

Unreasonable Results A commercial airplane has an air speed of m/s due east and flies with a strong tailwind. It travels km in a direction south of east in h.

(a) What was the velocity of the plane relative to the ground?

(b) Calculate the magnitude and direction of the tailwind’s velocity.

(c) What is unreasonable about both of these velocities?

(d) Which premise is unreasonable?

Suppose you take two steps A and B (that is, two nonzero displacements). Under what circumstances can you end up at your starting point? More generally, under what circumstances can two nonzero vectors add to give zero? Is the maximum distance you can end up from the starting point A+B the sum of the lengths of the two steps?

Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg \({\rm{B}}\), which is \(20.0\;{\rm{m}}\) in a direction exactly \(40^\circ \) south of west, and then leg \({\rm{A}}\), which is \(12.0\;{\rm{m}}\) in a direction exactly \(20^\circ \) west of north. (This problem shows that \({\rm{A}} + {\rm{B}} = {\rm{B}} + {\rm{A}}\).)

Construct Your Own Problem Consider an airplane headed for a runway in a cross wind. Construct a problem in which you calculate the angle the airplane must fly relative to the air mass in order to have a velocity parallel to the runway. Among the things to consider are the direction of the runway, the wind speed and direction (its velocity) and the speed of the plane relative to the air mass. Also calculate the speed of the airplane relative to the ground. Discuss any last minute maneuvers the pilot might have to perform in order for the plane to land with its wheels pointing straight down the runway.

Explain why it is not possible to add a scalar to a vector.

If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?

Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors \({\rm{A}}\), \({\rm{B}}\), and \({\rm{C}}\), all having different lengths and directions. Find the sum \({\rm{A}} + {\rm{B}} + {\rm{C}}\) then find their sum when added in a different order and show the result is the same. (There are five other orders in which \({\rm{A}}\), \({\rm{B}}\), and \({\rm{C}}\) can be added; choose only one.)

Suppose you add two vectors A and B. What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?

Show that the sum of the vectors discussed in Example 3.2 gives the result shown in Figure 3.24.