Section 4.4 Velocity as a Vector

Section 4.4 Velocity as a Vector In elementary problems, the speed of a moing object is calculated by diiding the distance traelled by the trael time. In adanced work, speed is defined more efully as the rate of change of distance with time. In any case, speed is a quantity haing magnitude only, so it is classified as a scalar. When the direction of motion as well as its magnitude is important, the correct term to use is elocity. Velocity is a ector quantity. Speed is the magnitude of a elocity. Velocity ectors can be added. When you walk forward in the aisle of an aircraft in flight, the 2-km/hr elocity of your walk adds to the 500-km/hr elocity of the plane, making your total elocity 502 km/hr. When two elocities are not in the same direction, the resultant elocity determined from the addition of two elocity ectors is neertheless a meaningful, physical quantity. EXAMPLE 1 A canoeist who can paddle at a speed of 5 km/h in still water wishes to cross a rier 400 m wide that has a current of 2 km/h. If he steers the canoe in a direction perpendicular to the current, determine the resultant elocity. Find the point on the opposite bank where the canoe touches. As the canoe moes through the water, it is ried sideways by the current. So een though its heading is straight across the current, its actual direction of motion is along a line angling downstream determined by the sum of the elocity ectors. From the ector diagram, 400 m current Vector Diagram 5 km/h x 2 km/h 2 (5) 2 (2) 2 and tan 2 5 29 5.4 km/h 21.8º Therefore, the canoeist crosses the rier at a speed of 5.4 km/h along a line at an angle of about 22º. The displacement triangle is similar to the ector triangle. 5 km/h 2 km/h 400 m x x 2 400 5 x 160 4.4 VELOCITY AS A VECTOR 145

He touches the opposite bank at a point 160 m downstream from the point directly opposite his starting point. We could also find x using the angle, but we must be eful not to round off in the process. EXAMPLE 2 Suppose the canoeist of Example 1 had wished to trael straight across the rier. Determine the direction he must head and the time it will take him to cross the rier. 400 m current In order to trael directly across the rier, the canoeist must steer the canoe slightly upstream. This time, it is the ector sum, not the heading of the canoe, which is perpendicular to the rier bank. From the ector diagram, Vector Diagram 5 km/h 2 km/h 2 (5) 2 (2) 2 and sin () 2 5 21 4.6 km/h 23.6º Therefore, to trael straight across the rier, the canoeist must head upstream at an angle of about 24º. His crossing speed will be about 4.6 km/h. The time it takes to cross the rier is calculated from rier width t cr ossing speed 0.4 21 0.087 h or 5.2 min (where the width is 0.4 km) (we aoid using rounded alues if possible) It takes the canoeist approximately 5.2 minutes to cross the rier. Wind affects a plane s speed and direction much the same way that current affects a boat s. The airspeed of a plane is the plane s speed relatie to the mass of air it is flying in. This may be different in both magnitude and direction from the plane s ground speed, depending on the strength and direction of the wind. EXAMPLE 3 An airplane heading northwest at 500 km/h encounters a wind of 120 km/h from 25º north of east. Determine the resultant ground elocity of the plane. Since the wind is blowing from 25º north of east, it can be represented by a ector whose direction is west 25º south. This wind will blow the plane off its course, 146 CHAPTER 4

changing both its ground speed and its heading. Let be the airspeed of the plane and w be the wind speed. On a set of directional axes, draw the two elocity ectors. Then draw the resultant elocity using the parallelogram law of ector addition. Plane Heading W Wind direction 45 In parallelogram OCBA, COA 45º 25º 70º, so OAB 110º. Then, in OAB, two sides and the included angle are known, so the magnitude of the resultant elocity can be calculated using the cosine law. w 2 500 2 120 2 2(500)(120) cos 110º w 552.7 Store this answer in your calculator memory. Next, AOB can be calculated from the sine law. N S sin AOB sin 110º (use the alue of 500 w + w calculated aboe) AOB 58.2º WOB 58.2º 25º 33.2º The resultant elocity has direction 33º north of west and a magnitude of 553 km/h. E B W C + w A w B N S O E 500 + w 110 O 120 A 110 45 45 65 25 A A key step in soling problems such as that in Example 3 is to find an angle in the triangle formed by the ectors. Here is a helpful hint: identify which angle is formed by ectors whose directions are gien, and draw small axes at the ertex of that angle. The diagram shows this alternate way to calculate that OAB 110º in Example 3. Vectors are needed to describe situations where two objects are moing relatie to one another. When astronauts want to dock the space shuttle with the international space station, they must match the elocities of the two craft. As they approach, astronauts on each spacecraft can picture themseles to be stationary and the other craft to be moing. When they finally dock, een though the two spacecraft are orbiting the earth at thousands of miles per hour, their relatie elocity is zero. 4.4 VELOCITY AS A VECTOR 147

Relatie elocity is the difference of two elocities. It is what an obserer measures, when he perceies himself to be stationary. The principle that all elocities are relatie was originally formulated by Einstein and became a cornerstone of his Theory of Relatiity. When two objects A and B hae elocities A and B,respectiely, the elocity of B relatie to A is rel B A EXAMPLE 4 A traelling east at 110 km/h passes a going in the opposite direction at 96 km/h. a. What is the elocity of the relatie to the? b. The turns onto a side road and heads northwest at the same speed. Now what is the elocity of the relatie to the? The ector diagram shows the elocity ectors of the and the. These elocities are relatie to someone standing by the side of the road, watching the two ehicles pass by. Since the is going east, let its elocity be 110. Then the s elocity is 96. rel (96) (110) 206 km/h or 206 km/h west rel This is the elocity that the appears to hae, according to the drier of the. b. After the turns, the angle between the and the elocities is 135º. The magnitude of the sum is found using the cosine law. rel 2 (96) 2 (110) 2 2(96)(110) cos 135º rel 190.4 km/h (Store this in your calculator.) The angle of the relatie elocity ector can be calculated from the sine law. si n 9 6 si n 135º 1 90.4 20.9º 135 rel 148 CHAPTER 4

After the turns, its elocity is 190 km/h in a direction W 21º N relatie to the. Note that the relatie elocity of the two ehicles does not depend on their position. It remains the same as long as the two ehicles continue to trael in the same directions without any changes in their elocities. Exercise 4.4 Part A Communication Knowledge/ Understanding 1. A plane is heading due east. Will its ground speed be greater than or less than its airspeed, and will its flight path be north or south of east when the wind is from a. N b. S 80º W c. S 30º E d. N 80º E 2. A man can swim 2 km/h in still water. Find at what angle to the bank he must head if he wishes to swim directly across a rier flowing at a speed of a. 1 km/h b. 4 km/h Knowledge/ Understanding 3. A street, a bus, and a taxi are traelling along a city street at speeds of 35, 42, and 50 km/h, respectiely. The street and the taxi are traelling north; the bus is traelling south. Find a. the elocity of the street relatie to the taxi b. the elocity of the street relatie to the bus c. the elocity of the taxi relatie to the bus d. the elocity of the bus relatie to the street Application Part B 4. A rier is 2 km wide and flows at 6 km/h. A motor boat that has a speed of 20 km/h in still water heads out from one bank perpendicular to the current. A marina lies directly across the rier on the opposite bank. a. How far downstream from the marina will the boat reach the other bank? b. How long will it take? 5. An airplane is headed north with a constant elocity of 450 km/h. The plane encounters a west wind blowing at 100 km/h. a. How far will the plane trael in 3 h? b. What is the direction of the plane? 6. A light plane is traelling at 175 km/h on a heading of N8º E in a 40-km/h wind from N80º E. Determine the plane s ground elocity. 4.4 VELOCITY AS A VECTOR 149

Application 7. A boat heads 15º west of north with a water speed of 3 m/s. Determine its elocity relatie to the ground when there is a 2 m/s current from 40º east of north. 8. A plane is steering east at a speed of 240 km/h. What is the ground speed of the plane if the wind is from the northwest at 65 km/h? What is the plane s actual direction? 9. Upon reaching a speed of 215 km/h on the runway, a jet raises its nose to an angle of 18º with the horizontal and begins to lift off the ground. a. Calculate the horizontal and ertical components of its elocity at this moment. b. What is the physical interpretation of each of these components of the jet s elocity? 10. A pilot wishes to fly to an airfield S20º E of his present position. If the aerage airspeed of the plane is 520 km/h and the wind is from N80º E at 46 km/h, a. in what direction should the pilot steer? b. what will the plane s ground speed be? 11. A destroyer detects a submarine 8 nautical miles due east traelling northeast at 20 knots. If the destroyer has a top speed of 30 knots, at what heading should it trael to intercept the submarine? Part C 12. An airplane flies from to Toronto to Vancouer and back. Determine which time is shorter. a. The time for the round trip when there is a constant wind blowing from Vancouer to Toronto. b. The time for the round trip when there is no wind. 13. A sailor climbs a mast at 0.5 m/s on a ship traelling north at 12 m/s, while the current flows east at 3 m/s. What is the speed of the sailor relatie to the ocean floor? 14. A is 260 m north and a 170 m west of an intersection. They are both approaching the intersection, the from the north at 80 km/h, and the from the west at 50 km/h. Determine the elocity of the relatie to the. 150 CHAPTER 4