Chapter 3

1. Chapter 3 Projectile Motion Alta High Conceptual Physics

2. Vectors and Scalars • Vector vs. Scalars • Vectors have both magnitude and direction • Displacement, Velocity, Acceleration, Force, and Momentum • Scalars have only magnitude • Mass, Time, and Temperature Alta High Conceptual Physics

3. Vector Addition • Two Methods • Graphical Method • Requires a ruler and a protractor • Process • Convert each vector into a line that fits a scale of your choosing • Draw Vectors head to tail, measuring the angle exactly • Draw a resultant • Measure the resultant and the resulting angle • Convert the measurement back into a vector answer Alta High Conceptual Physics

4. Graphical Addition • A man drives 125 km west and then turns 45º North of West and travels an additional 100 km. What is his displacement? • Step one: Set Scale • 1 cm = 25 Km 100 km = 4 cm 45° Alta High Conceptual Physics 125 km = 5 cm

5. Graphical Addition Step Two: Move Vectors Head to Tail Step Three: Draw Resultant from Tail of First Vector to the head of the last. 100 km = 4 cm 45° 125 km = 5 cm Alta High Conceptual Physics

6. Graphical Addition Step Four: Measure the Measure the Resultant and the resulting angle. Step Five: Using the Scale, convert the measurement into an answer. 100 km = 4 cm 8.3 cm 8.3 cm * 25 Km/cm = 208 Km 45° 125 km = 5 cm 26.2° Final Answer: His displacement is 208 Km @ 19.9° North of West Alta High Conceptual Physics

7. Components of Vectors • Often we need to change a single vector into two vectors in order to make addition easier. These two vectors will be a right angles to each other and are called component vectors or just components for short. • The breaking down of a vector this way is called resolution. Alta High Conceptual Physics

8. Vector Resolution Step One: Draw Original Vector 45º 100 Km 45° North of West Alta High Conceptual Physics

9. Vector Resolution Step Two: Draw parallelogram to Resolve Vector into components 100 Km 45° North of West Alta High Conceptual Physics

10. Vector Resolution Step Three: Label and measure components North component = 70.7 km 45° West component = 70.7 km 100 Km 45° North of West Alta High Conceptual Physics

11. Mathematical Addition • Mathematical Addition of Vectors Requires a basic knowledge of Geometry – You must know: • Sin θ = Opposite/Hypotenuse • Cos θ = Adjacent/Hypotenuse • Tan θ = Sin θ/ Cos θ =Opposite/Adjacent • ArcSin, ArcCos, ArcTan Alta High Conceptual Physics

12. Mathematical Addition • Mathematical addition requires that you be able to • Draw a rough sketch of the original vectors • Draw a parallelogram • Draw the Resultant • Use Triangle geometry to find the magnitude and direction of the resultant Alta High Conceptual Physics

13. Example What is the result of adding two vectors if the first vector is 100 km to the west and the second vector is 50 km to the north? Alta High Conceptual Physics

14. Mathematical Addition Step One: Sketch the vectors Step Two: Put vectors on the same axis and draw the parallelogram + 100 Km West 50 Km North Step One Step Two Alta High Conceptual Physics

15. Mathematical Addition Step Three: Draw the Resultant Step 4: Use Right triangle geometry to find the length of the resultant Step Four: Since this is a right triangle you can find the resultant using Pythagorean theorem Pythagorean Theorem: C2 = A2 + B2 C = SQRT(A2 + B2) = SQRT{(100)2+(50)2} =111.8 Km 50 Km 100 Km 111.8 Km Step Three Alta High Conceptual Physics

16. Graphical vs. Mathematical Methods • Graphical Method • Less Complicated Math • Good Approximation • Not as exact as you would like • Mathematical Method • No need for ruler or protractor • Less time dedicated to sketch • More Accurate result • Mathematics is more difficult Alta High Conceptual Physics

17. Projectile Motion • A projectile is any object thrown or launched • Uses the same kinematic equations you have already learned • Requires that you use two sets of equations; one in the horizontal and one in the vertical • These two sets are related only by time Alta High Conceptual Physics

18. Projectile Motion • A projectile which is dropped or thrown horizontally has an initial velocity in the Y direction (V0y) = 0 • Acceleration in the horizontal direction (ax) = 0; hence Velocity in the horizontal direction (Vx) is constant • This means that only one kinematic equation applies in the horizontal direction: Xx = Vxt Alta High Conceptual Physics

19. Projectile Motion • While it is moving forward, it is also falling just as if you had dropped it. • So in the Y direction: x = ½ at2 • Where a is the acceleration due to gravity. • The one thing that both motions have in common is time. When the projectile hits the ground it stops moving in both directions! Alta High Conceptual Physics

20. Projectile Motion A ball is thrown horizontally off a building which is 200 m high with a velocity of 10 m/s • How long does it take to reach the ground? • How far from the building will it land? • What is its velocity just before it hits the ground (remember magnitude and direction) Alta High Conceptual Physics

21. Fast Moving Projectiles: Satellites • A satellite is a fast moving projectile • Since the projectile moves in a curved path, and the earth is a curved surface, if the projectile moved fast enough its motion would follow the curve of the earth • Such a projectile would fall around the earth rather than to the earth • Such a projectile becomes a satellite and orbits the earth. Alta High Conceptual Physics

22. Problem Types • Adding Vectors • Resolving vectors into components • Projectile Motion Alta High Conceptual Physics