2. Today, we will have a short review on vectors and projectiles and then have a quiz. You will need a calculator, a clicker and some scratch paper for the quiz.

3. Vectors Vector: a quantity that has both magnitude (size) and direction Examples: displacement, velocity, acceleration Scalar: a quantity that has no direction associated with it, only a magnitude Examples: distance, speed, time, mass

4. Vectors are represented by arrows. The length of the arrow represents the magnitude (size) of the vector. And, the arrow points in the appropriate direction. 20 m/s 50 m/s NW East

5. Adding vectors graphically 1.Without changing the length or the direction of any vector, slide the tail of the second vector to the tip of the first vector. 2. Draw another vector, called the RESULTANT, which begins at the tail of the first vector and points to the tip of the last vector. +

6. Subtracting vectors graphically 1.First, reverse the direction of the vector you are subtracting. Then, without changing the length or the direction of any vector, slide the tail of the second vector to the tip of the first vector. 2. Draw another vector, called the RESULTANT, which begins at the tail of the first vector and points to the tip of the last vector.

7. Adding co-linear vectors (along the same line) A = 8 m B = 4 m A + B = R = 12 m C = 10 m/s D = - 3 m/s C + D = 10 + (-3) = R = 7 m/s

8. Airplane Tailwind

9. Airplane Headwind

10. Adding perpendicular vectors How could you find out the length of the RESULTANT? Since the vectors form a right triangle, use the PYTHAGOREAN THEOREM A 2 + B 2 = C 2 10 m 6 m 11.67 m

12. A boat capable of going 3 m/s in still water is crossing a river with a current of 5 m/s. If the boat points straight across the river, where will it end up- straight across the river or downstream? Downstream, because that is where the current will carry it even as it goes across! What is the resultant velocity?

13. Resultant velocity Sketch the 2 velocity vectors: the boat’s velocity, 5 m/s and the water’s velocity, 3 m/s Resultant velocity, V b = 3 m/s V w = 5 m/s Resultant velocity

15. Vector COMPONENTS Each vector can be described to terms of its x and y components. X (horizontal) component Y (vertical) component If you know the lengths of the x and y components, you can calculate the length of the vector using the Pythagorean.

17. Drawing the x and y components of a vector is called “resolving a vector into its components” Make a coordinate system and slide the tail of the vector to the origin. Draw a line from the arrow tip to the x-axis. The components may be negative or positive or zero. X component Y component

18. Calculating the components How to find the length of the components if you know the magnitude and direction of the vector. Sin  = opp / hyp Cos  adj / hyp Tan  = opp / adj SOHCAHTOA A AxAx AyAy  = A sin  = A cos  = 12 m/s = 35 degrees = 12 cos 35 = 9.83 m/s = 12 sin 35 = 6.88 m/s

19. Are these components positive or negative?

20. V = 22 m/s  = 50˚ What is v x ? V x = - v cos  ˚ V x = -22 cos 50˚ V x = - 14.14 m/s What is v y ? V y = v sin  ˚ V y = 22 sin 50˚ V y = 16.85 m/s

21. Finding the angle Suppose a displacement vector has an x-component of 5 m and a y-component of - 8 m. What angle does this vector make with the x-axis?  = ? We are given the side adjacent to the angle and the side opposite the angle. Which trig function could be used? Tangent  = opposite ÷ adjacent Use “arc tangent” to determine the angle “tan -1 ” Therefore the angle  = tan -1 (8 ÷ 5)  = 57.99 degrees below the positive x-axis

23. Adding Vectors by components Slide each vector to the origin. Resolve each vector into its x and y components The sum of all x components is the x component of the RESULTANT. The sum of all y components is the y component of the RESULTANT. Using the components, draw the RESULTANT. Use Pythagorean to find the magnitude of the RESULTANT. Use inverse tan to determine the angle with the x-axis.   A B R x y A = 18,  degrees B = 15,  = 40 degrees 18 cos 20 18 sin 20 -15 cos 4015 sin 40 5.42 15.8  tan -1 (15.8 / 5.42) = 71.1 degrees above the positive x-axis  A B

24. Walking on a train- relative velocity What is the velocity of the man on the train relative to the person standing outside on the ground?

25. A boat capable of going 5 m/s in still water is crossing a river with a current of 3 m/s. If the boat points straight across the river, where will it end up- straight across the river or downstream? Downstream, because that is where the current will carry it even as it goes across! What is the resultant velocity?

26. Resultant velocity Sketch the 2 velocity vectors: the boat’s velocity, 5 m/s and the river’s velocity, 3 m/s Resultant velocity, 5 m/s 3 m/s Resultant velocity