Riverboat and Airplane Vectors

Transcription

1 Grade Homework Riverboat and Airplane Vectors It all depends on your point of view It s all relative On occasion objects move within a medium that is moving with respect to an observer. In such instances, the magnitude of the velocity of the moving object will not be same as the speedometer reading of the vehicle. The speedometer might read 20km/hr, but the motorboat might be moving 25km/hr relative to an observer on the shore. Making up time on an airplane Making up time on an airplane 1

2 Making up time on an airplane Losing time in an airplane Losing time in an airplane Losing time in an airplane Crosswinds Now consider a plane travelling 100km/hr, South that encounters a crosswind of 25km/hr Crosswinds Now consider a plane travelling 100km/hr, South that encounters a crosswind of 25km/hr (100 km/hr) 2 + (25 km/hr) 2 = R km 2 /hr km 2 /hr 2 = R km 2 /hr 2 = R 2 SQRT( km 2 /hr 2 ) = R km/hr = R 2

3 Which direction did it go and at what angle? Using the Pythagorean Theorem we know the Resultant Velocity, but we need to use another tool to solve for the direction the plane went off course The direction of a resultant vector can be determined by using the trigonometric functions SOHCAHTOA SOH stands for Sine equals Opposite over Hypotenuse CAH stands for Cosine equals Adjacent over Hypotenuse TOA stands for Tangent equals Opposite over Adjacent With the Hiker problem from before, we can determine the direction of the hiker s overall displacement The measure of the angle as determined by SOHCAHTOA is not always the direction of the vector Make sure that you are following the direction in which the vectors are travelling 3

4 Practice Problem Practice Problem tan(theta) = (5/10) = 0.5 Theta = tan-1 (0.5) Theta = 26.6 degrees Direction of R = 90 deg deg Direction of R = deg Practice Problem Crosswinds tan(theta) = (40/30) = Theta = tan-1 (1.333) Theta = 53.1 degrees Direction of R = 180 deg deg Direction of R = deg The direction of the resulting velocity can be determined using a trigonometric function. tan (Θ) = (opposite/adjacent) tan (Θ) = (25/100) Θ= -tan (25/100) Θ = 14.0 degrees If the resultant velocity of the plane makes a 14 angle with a southward direction, then the resultant is 256 Tailwind Headwind Crosswind The affect of wind on an airplane is similar to the affect of river current upon a motorboat. If a motorboat were to head straight across a river, it would not reach the shore directly across from its starting point. 4

5 Motorboat questions typically are accompanied by three separate questions 1. What is the resultant velocity (both magnitude and direction) of the boat? 2. If the width of the river is x meters wide, then how much time does it take the boat to The first question can be answered using the same method we used on the airplanes. 1. What is the resultant velocity (both magnitude and direction) of the boat? The 2 nd and 3 rd of these questions can be answered using the average speed equation (and a lot of logic) 2. If the width of the river is x meters wide, then how much time does it take the boat to travel shore to shore? reach the opposite shore? A motorboat traveling 4m/s E encounters a current traveling 3.0m/s N. 1. What is the resultant velocity of the motorboat? (4.0 m/s) 2 + (3.0 m/s) 2 = R 2 16 m 2 /s m 2 /s 2 = R 2 4m/s 3m/s 25 m 2 /s 2 = R 2 SQRT (25 m 2 /s 2 ) = R 5.0 m/s = R tan (Θ) = (opposite/adjacent) tan (Θ) = (3/4) Θ = -tan (3/4) Θ = 36.9 degrees 2. If the width of the river is 80 meters wide, time = distance / average speed Which average speed do we use? 2. If the width of the river is 80 meters wide, time = distance / average speed Be careful and pay attention to which distance you have been given 2. If the width of the river is 80 meters wide, time = distance / average speed time = 80m / 4m/s time = 20s 5

6 distance = time * average speed Again, we have to determine which of the three average speeds to use distance = time * average speed If the boat is traveling downstream with the current, then we have to use river speed Go to the Boards distance = time * average speed distance = 3m/s * 20s distance=60m A plane can travel with a speed of 80 mi/hr with respect to the air. Determine the resultant velocity of the plane (magnitude only) if it encounters a a. 10 mi/hr headwind. b. 10 mi/hr tailwind. c. 10 mi/hr crosswind. d. 60 mi/hr crosswind. Go to the Boards A motorboat traveling 5 m/s, East encounters a current traveling 2.5 m/s, North. a. What is the resultant velocity and direction of the motor boat? b. If the width of the river is 80 meters wide, c. What distance downstream does the boat Homework Riverboat and Airplane Next class-projectile Motion 6