SPH3U1 Lesson 05 Kinematics

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1 VECTORS IN TWO-DIMENSIONS LEARNING GOALS Students will Draw vector scale diagrams to visualize and analyze the nature of motion in a plane. Analyze motion by using scale diagrams to add vectors. Solve two-dimensional vector problems using scale diagrams and trigonometry. WEBSITE AND TEXTBOOK RESOURCES Reading Video Nelson Physics 11 Section 2.1 Pg Physics Classroom Vectors Khan Academy (Visualizing 2D Vectors) Earl Haig (Putting Bearings on Vectors and Adding Vectors) Interactive Figures Adding and Subtracting Vectors (Visual Demonstrations) DISPLACEMENT VECTORS Balls, rockets, bullets, cars, boats, planes, and sparks all have 2D motion. We can quantify their motion using horizontal and vertical vectors. We have seen both horizontal and vertical vectors of objects in motion. A. Draw a displacement vector that represents an object moving in the East direction and a separate vector representing an object moving in the North direction. Indicate the initial and final positions of the object in both cases. The displacement, Δ of an object is a vector that points from an initial position, to its final position,. The vector s magnitude is equal to the straight-line distance between the two positions. STEPS TO DRAWING VECTORS 1. Draw an x,y axis. 2. Measure the angle starting from the x-axis or based on the letter direction provided. A direction like [N15 0 E] is read north, fifteen degrees east. This direction is found by starting at the north line and measuring 15 0 towards the east. 3. Draw the vector based on a length scale you specify. W N S 15 0 [N15 0 E] E MAGNITUDE AND DIRECTION The following diagram depicts the motion of a typical student. Based on the displacement vectors shown in the diagram, what is the magnitude and direction of the student s displacement? (Hint: Start with a protractor located at the student s home to help you find the direction.) 1

From Home to School From Home to the Diner From Home to the Sports Complex VECTORS IN A PLANE Imagine describing the motion of an expert water-skier to someone who had not watched the skier

You will represent vector quantities with arrows that point in the direction of the quantity.

2 From Home to School From Home to the Diner From Home to the Sports Complex VECTORS IN A PLANE Imagine describing the motion of an expert water-skier to someone who had not watched the skier demonstrate his technique. You would probably do a lot of pointing in different directions. In a sense, you would be using vectors to describe the skier s motion. You will represent vector quantities with arrows that point in the direction of the quantity. The length of the arrow is proportional to the magnitude of the quantity you are representing, so you need an appropriate scale to represent the magnitude. Vector quantities have direction, so you need a frame of reference or coordinate system to represent a direction. The map to the right is an excellent example of using vectors to locate the displacement between two objects. Each displacement vector has an appropriate length that follows the scale at the bottom and the angle drawn correctly. ADDING VECTORS GRAPHICALLY Addition of vectors starts with some basic rules of arithmetic and then includes a few more rules. You have known for a long time that you cannot add apples and oranges or centimetres and metres. Similarly, you can only add vectors that represent the same quantity and are drawn to the same scale. Read the steps on the next page to find out how to add vectors graphically. Then use the method described to do EXAMPLE 1 below. 2

Find the total displacement of the dog by drawing a scale vector

3 EXAMPLE 1: ADDING GRAPHICALLY A dog chases a cat 400 m [W] and turns to travel another 600 m [N50 o E]. Find the total displacement of the dog by drawing a scale vector diagram. Check your answer using sine/cosine laws on the triangle you drew. is the displacement Δ! 3

4 PRACTICE PROBLEMS 1. An airplane flies 100 km north in 20 min, then 150 km west in 40 min and finally 500 km south in 52 min. a. What is the total distance and total displacement of the plane? b. What are the average velocity and average speed for the trip? 2. A pleasure boat heads out from the marina and travels 2.7 km [S] to a small island. It then travels 3.4 km [S26 0 E] to another island. What is the boats displacement for the entire journey? Solve using a scale diagram and then solve again using the sine and cosine law. 3. A plane is flying at a constant velocity through the air at 120 km/h [S]. The wind is blowing at a velocity of 50 km/h [W]. These two velocities can be added to determine the plane s total velocity with respect to the ground. a. Find the total velocity (add the velocity vectors ie. tip-to-tail). b. If the plane is flying between two cities that are 500 km apart, how long does the trip take? 4. A jogger runs at a velocity of 2.8 m/s [W] for 50 minutes and then at a velocity of 3.2 m/s [N30 0 W] for 30 minutes. Calculate the total displacement of the jogger. 5. A hiker heads [N40 0 W] for 4.0 km, then [E10 0 N] for 3.0 km and finally she walks 2.5 km [S40 0 W]. (Hint: this one has to be done in 2 steps if you use sine and cosine law, but can be done in one step using a scale diagram.) a. What was her total displacement? b. In what direction should she walk to get back to her starting point in the shortest possible distance? c. If her average speed is 4.0 km/h for the entire trip, what was her average velocity? Answers: 1. a) 750 km and 427 km [S21 0 W] b) 229 km/h [S21 0 W] km [S15 0 E] 3. a) 130 km/h [S23 0 W] b) 3.8 h x 10 4 m [W24 0 N] 5. a) 2.1 km [N36 0 W] b) [S36 0 E] c) 0.87 km/h [N36 0 W] 4

5 VECTORS IN TWO-DIMENSIONS VECTOR COMPONENTS LEARNING GOALS Students will use components to add vectors by: Splitting vectors into simpler horizontal and vertical parts; Adding the vertical parts to each other, then adding the horizontal parts to each other; Finally adding the horizontal and vertical parts to each other. WEBSITE AND TEXTBOOK RESOURCES Reading Nelson Physics 11 Section 2.2 Pg Physics Classroom Vector Components and Addition Interactive Figure Vector Addition of Components ADDING SCALARS SIMPLY Say you want to add in your head. You might first split them up: 67 = Then you might add = 140 Then you might add = 12 Finally you add = = You often use this method in your head because it is easier to do each simpler operation than to do the entire addition at once. The component method for adding vectors works the same way. Each vector is split into simpler parts we call components that are easier to add. VECTOR COMPONENTS Vectors in 2-D have their direction often in between two simple, cardinal directions; for example 34 m [N32 0 E] is between north and east. This has a north part and an east part (see diagram) 45 m [W15 0 S] is between west and south. This has a west part and a south part (see diagram) N x 32 = 34 = = = 34 = 3432 = 28.8 y 34 m 32 0 W y x 45 m 15 0 E 15 = 45 = = = S Study the calculations in the diagram. Each of the two vectors has been split up using simple trigonometry into two other vectors (dashed lines, placed tip-to-tail) that add up to the original vector. These new vectors are called components. They are all parallel to the axes. 5

6 Instead of adding the two original vectors, we now add up the four components instead: 18.0 m [E] m [N] m [W] m [S] First, add the N-S (or y) vectors: 28.8 m [N] m [S] = 17.2 m [N] Then, add the E-W (or x) vectors: 18.0 m [E] m [W] = 25.5 m [W] You now have the components of the answer. Add them using a scale diagram or using the Pythagorean theorem and the tangent of an angle: N = = m R!" = " = 34 W 25.5 m θ E The sum of the two vectors is therefore 30.8 m [W34 0 N]. S This method takes about as much math as using the sine and cosine laws to add the vectors; however, in this case, all the trigonometry you need to know is SOH CAH TOA. If you have to add three or more vectors, this method is MUCH FASTER than using the sine and cosine laws. Any number of vectors can be added together at once. PRACTICE PROBLEMS Pg 69 Q1,2 Pg 71 Q1,2Pg. 75 Q1-6 6

SPH3U1 Lesson 09 Kinematics

SPH3U1 Lesson 09 Kinematics VECTORS IN TWO-DIMENSIONS LEARNING GOALS Students will Draw vector scale diagrams to visualize and analyze the nature of motion in a plane. Analyze motion by using scale diagrams to add vectors. Solve