Two Dimensional Motion and Vectors

1. Physics – Mrs. Dimler Two Dimensional Motion and Vectors

2. A scalar is a physical quantity that has magnitude (size) but no direction. Examples: • Temperature • Mass • Time • Volume • Speed • Distance Scalar Quantities

3. A vector is a physical quantity that has both magnitude and direction. Examples: • Displacement • Velocity • Acceleration • Force • Momentum Vector quantities

4. Why am I learning about vectors?? Velocity vectors associated with projectile motion

5. Vector components

6. If two vectors have unequal magnitudes, can their sum be zero? • Is it possible to add a vector quantity to a scalar quantity? Explain. • Could you add a velocity vector to a displacement vector? • Could you add two displacement vectors that are not expressed in the same units? • Which of the following quantities are scalars, and which are vectors? • The acceleration of a plane as it takes off • The number of passengers on the plane • The duration of the flight • The displacement of the flight • The amount of fuel required for the flight Check your conceptual understanding

7. Scalar – a physical quantity that has magnitude but no direction Vector – a physical quantity that has both magnitude and direction Resultant – a vector that represents the sum of two or more vectors Terms to Know

8. A scalar is a quantity completely specified by only a number with appropriate units, whereas a vector is a quantity that has magnitude and direction. Vectors can be added graphically using the triangle method of addition, in which the tail of one vector is placed at the head of the other. The resultant is the vector drawn from the tail of the first vector to the head of the last vector. Introduction to vectors – Key ideas

9. http://www.khanacademy.org/science/physics/mechanics/v/introduction-to-vectors-and-scalarshttp://www.khanacademy.org/science/physics/mechanics/v/introduction-to-vectors-and-scalars Khan academy intro to vectors and scalars

10. http://www.youtube.com/watch?v=A05n32Bl0aY Despicable Me – Vector introduction

11. One method for diagraming the motion of an object employs vectors and the use of the x- and y-axes. • Axes are often designated using fixed directions. • In the figure shown here, the positive y-axis points north and the positive x-axis points east. Coordinate System in Two Dimensions

12. The Pythagorean Theorem • Use the Pythagorean theorem to find the magnitude of the resultant vector. • The Pythagorean theorem states that for any right triangle, the square of the hypotenuse—the side opposite the right angle—equals the sum of the squares of the other two sides, or legs. Determining resultantMagnitude and direction cont.

13. Hypotenuse Opposite Adjacent Trig functions θ SOH CAH TOA

14. http://www.khanacademy.org/science/physics/mechanics/v/optimal-angle-for-a-projectile-part-1http://www.khanacademy.org/science/physics/mechanics/v/optimal-angle-for-a-projectile-part-1 Khan Academy VideoVector components of projectile motion

15. Find the components of the velocity of a helicopter traveling 95 km/h at an angle of 35° to the ground. Leave answer in units of km/h. A truck drives up a hill with a 15° incline. If the truck has a constant speed of 22 m/s, what are the horizontal and vertical components of the truck’s velocity? White Board problems

16. vy = 54 km/h; vx = 78 km/h Vx = 21 m/s, vy = 5.7 m/s ANSWERS

17. What are the horizontal and vertical components of a cat’s displacement when the cat has climbed 5m directly up a tree? Answer: dx = 0m, dy = 5m Conceptual challenge

18. An LSU football player (GEAUX TIGERS) runs directly down the field for 35m before turning to the right at an angle of 25° from his original direction and running an additional 15m before getting tackled. What is the magnitude and direction of the runner’s total displacement? Student White board problem

19. Total displacement = 49 m at 7.3° to the right of downfield Answers

20. http://www.khanacademy.org/science/physics/mechanics/v/optimal-angle-for-a-projectile-part-2---hangtimehttp://www.khanacademy.org/science/physics/mechanics/v/optimal-angle-for-a-projectile-part-2---hangtime http://www.khanacademy.org/science/physics/mechanics/v/optimal-angle-for-a-projectile-part-3---horizontal-distance-as-a-function-of-angle--and-speed Khan academy videos – Projectile motion

21. Objects that are thrown or launched into the air and are subject to gravity are called projectiles. Projectile Motion is the curved path that an object follows when thrown, launched, or otherwise projected near the surface of the earth. The path of a projectile is a curve called a parabola.If air resistance is disregarded, projectiles follow parabolic trajectories. Projectile motion is free fall with an initial horizontal velocity. Projectile motion

22. Projectile Motion • The yellow ball is given an initial horizontal velocity and the red ball is dropped. Both balls fall at the same rate. • In this book, the horizontal velocity of a projectile will be considered constant. This would not be the case if we accounted for air resistance.

23. How can you know the displacement, velocity, and acceleration of a projectile at any point in time during its flight? • One method is to resolve vectors into components, then apply the simpler one-dimensional forms of the equations for each component. • Finally, you can recombine the components to determine the resultant Kinematic equations for projectiles

24. To solve projectile problems, apply the kinematic equations in the horizontal and vertical directions. • In the vertical direction, the acceleration ay will equal –g (–9.81 m/s2) because the only vertical component of acceleration is free-fall acceleration. • In the horizontal direction, the acceleration is zero, so the velocity is constant. Kinematic equations for projectiles

25. Projectiles Launched Horizontally • The initial vertical velocity is 0. • The initial horizontal velocity is the total initial velocity. Projectiles Launched At An Angle • Resolve the initial velocity into x and y components. • The initial vertical velocity is the y component. • The initial horizontal velocity is the x component. Kinematic equations for projectiles

26. The Royal Gorge Bridge in Colorado rises 321m above the Arkansas River. Suppose you kick a rock horizontally off the bridge. The magnitude of the rock’s horizontal displacement is 45.0m. Find the speed at which the rock was kicked. Projectile Motion Problem

27. A Pelican flying along a horizontal path drops a fish from a height of 5.4 m. The fish travels 8.0m horizontally before it hits the water below. What is the pelican’s speed? If the pelican in the above problem was traveling at the same speed but was only 2.7m above water, how far would the fish travel horizontally before hitting the water below. Practice Problems

28. vx = 7.6 m/s d = 5.6 m Answers

29. If you are moving at 80 km/h north and a sports car passes you going 90 km/h, how fast would the other car seem to be going relative to you? How fast would the sports car be moving relative to a stationary observer? Relative Motion question

30. If you are moving at 80 km/h north and a car passes you going 90 km/h, to you the faster car seems to be moving north at 10 km/h. Someone standing on the side of the road would measure the velocity of the faster car as 90 km/h toward the north. This simple example demonstrates that velocity measurements depend on the frame of reference of the observer. Relative Motion – Frames of Reference

31. Relative Motion – Frames of reference Consider a stunt dummy dropped from a plane. (a) When viewed from the plane, the stunt dummy falls straight down. (b) When viewed from a stationary position on the ground, the stunt dummy follows a parabolic projectile path.

32. Introduction to Vectors • A scalar is a quantity completely specified by only a number with appropriate units, whereas a vector is a quantity that has both magnitude and direction. • Vector Operations • The Pythagorean theorem and the inverse tangent function can be used to find the magnitude and direction of a resultant vector. • Any vector can be resolved into its component vectors by using the sine and cosine functions. • Projectile Motion • Neglecting air resistance, a projectile has a constant horizontal velocity and a constant downward free-fall acceleration. • In the absence of air resistance, projectiles follow a parabolic path. • Relative Motion • Velocity measurements depend on the frame of reference of the observer. Key Ideas: