Physics Intro & Kinematics

1. Physics Intro & Kinematics • Quantities • Units • Vectors • Displacement • Velocity • Acceleration • Kinematics • Graphing Motion in 1-D

2. Some Physics Quantities Vector - quantity with both magnitude (size) and direction Scalar - quantity with magnitude only • Vectors: • Displacement • Velocity • Acceleration • Momentum • Force • Scalars: • Distance • Speed • Time • Mass • Energy

3. Mass vs. Weight • Mass • Scalar (no direction) • Measures the amount of matter in an object • Weight • Vector (points toward center of Earth) • Force of gravity on an object On the moon, your mass would be the same, but the magnitude of your weight would be less.

4. The length of the arrow represents the magnitude (how far, how fast, how strong, etc, depending on the type of vector). The arrow points in the directions of the force, motion, displacement, etc. It is often specified by an angle. Vectors Vectors are represented with arrows 5 m/s 42°

5. Vectors vs. Scalars • One of the numbers below does not fit in the group: • 35 ft • 161 mph • 70° F • 200-m, 30° East of North • 12 200 people

6. Vectors vs. Scalars The answer is: 200-m, 30° East of North Why is it different? • Numbers w/ magnitudeonly are called SCALARS. • Numbers w/ magnitude and direction are called VECTORS.

7. Vector Example • Particle travels from A to B along the path shown by the dotted red line • This is the distance traveled and is a scalar • The displacement is the solid line from A to B • displacement is independent of path taken between two points • displacement is a vector • Notice the arrow indicating direction

8. Other Examples of Vectors • Displacement (of 3.5 km at 20o North of East) • Velocity (of 50 km/h due North) • Acceleration (of 9.81 m/s2downward) • Force (of 10 Newtons in the +x direction)

9. Notation • Vectors are written as arrows. • length describes magnitude • direction indicates the direction of vector… • Vectors are written in bold text in your book • Conventions for written notation shown below…

10. Adding Vectors Case 1: Collinear Vectors

11. What is theground speedof an airplane flying with anair speed of 100 mph into aheadwind of 100 mph?

12. Adding Collinear Vectors When vectors are parallel: just add magnitudes and keep the direction. Ex: 50 mph east + 40 mph east = 90 mph east

13. Adding Collinear Vectors When vectors are antiparallel: just subtract smaller magnitude from larger - use the direction of the larger. Ex: 50 mph east + 40 mph west = 10 mph east

14. Adding Perpendicular Vectors When vectors are perpendicular: • sketch the vectors in a HEAD TO TAIL orientation • use right triangle trig to solve for the resultant and direction. Ex: 50 mph east + 40 mph south = ??

15. An Airplane flies north with an airspeed of 650 mph. If the wind is blowing east at 50mph, what is the speed of the plane as measured from the ground?

16. Adding Perpendicular Vectors 50 mph R 650 mph θ

17. Examples Ex1: Find the sum of the forces of 30 lb south and 60 lb east. Ex2: What is the ground speed of a speed boat crossing a river of 5mph current if the boat can move 20mph in still water?

18. An airplane flies north with an airspeed of 575 mph. 1. If the wind is blowing 30° north of east at 50 mph, what is the speed of the plane as measured from the ground? 2. What if the wind blew south of west?

19. Adding Skew Vectors When vectors are not collinear and not perpendicular, we must resort to drawing a scale diagram. • Choose a scale and a indicate a compass • Draw the vectors Head to Tail • Draw the resultant • Measure the resultant and the angle! Ex: 50 mph east + 40 mph south = ??

20. Adding Skew Vectors R θ Measure R with a ruler and measure θ with a protractor.

21. Vector Components • Vectors are described using their components. • Components of a vector are 2 perpendicular vectors that would add together to yield the original vector. • Components are notated using subscripts. F Fy Fx

22. Adding Vectors by Components A B

23. Adding Vectors by Components B A Transform vectors so they are head-to-tail.

24. Adding Vectors by Components Bx By B A Ay Ax Draw components of each vector...

25. Adding Vectors by Components B A By Ay Ax Bx Add components as collinear vectors!

26. Adding Vectors by Components B A By Ay Ry Ax Bx Rx Draw resultants in each direction...

27. Adding Vectors by Components B A R Ry q Rx Combine components of answer using the head to tail method...

28. Adding Vectors Graphically When you have many vectors, just keep repeating the process until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector

29. B B A A D D C C Adding Vectors Graphically, final Example: A car travels 3 km North, then 2 km Northeast, then 4 km West, and finally 3 km Southeast. What is the resultant displacement? R R is ~2.4 km, 13.5° W of N or 103.5º from +ve x-axis.

30. A car travels 3 km North, then 2 km Northeast, then 4 km West, and finally 3 km Southeast. What is the resultant displacement? Use the component method of vector addition. X-components Ax = 0 km Bx = (2 km) cos 45º = 1.4 km Cx= -4 km Dx= (3km) cos 45º = 2.1 km Y-components Ay = 3 km By = (2 km) sin 45º = 1.4 km Cy = 0 km Dy= (3km) sin 315º = -2.1 km Components of a Vector N y B A C By x Bx W E Dx Dy D S

31. Components of a Vector Rx = Ax + Bx + Cx + Dx = 0 km + 1.4 km - 4.0 km + 2.1 km = -0.5 km Ry = Ay + By + Cy + Dy = 3.0 km + 1.4 km + 0 km - 2.1 km = 2.3 km Magnitude: N y R Direction: Ry x E W Rx Stop. Think. Is this reasonable? NO! Off by 180º. Answer: -78º + 180° = 102° S

32. Adding Vectors by Components

33. Components of a Vector • The x-component of a vector is the projection along the x-axis: • The y-component of a vector is the projection along the y-axis:

34. Adding Vectors by Components Use the Pythagorean Theorem and Right Triangle Trig to solve for R and θ…

35. Quantity . . . Unit (symbol) Displacement & Distance . . . meter (m) Time . . . second (s) Velocity & Speed . . . (m/s) Acceleration . . . (m/s2) Mass . . . kilogram (kg) Momentum . . . (kg·m/s) Force . . .Newton (N) Energy . . . Joule (J) Units Units are not the same as quantities!

36. SI Prefixes Little Guys Big Guys

37. Kinematics – branch of physics; study of motion Position (x) – where you are located Distance (d) – how far you have traveled, regardless of direction Displacement (x) – where you are in relation to where you started Kinematics definitions

38. You drive the path, and your odometer goes up by 8 miles (your distance). Your displacement is the shorter directed distance from start to stop (green arrow). What if you drove in a circle? Distance vs. Displacement start stop

39. Speed, Velocity, & Acceleration • Speed (v) – how fast you go • Velocity (v) – how fast and which way; the rate at which position changes • Average speed ( v ) – distance/time • Acceleration (a) – how fast you speed up, slow down, or change direction; the rate at which velocity changes

40. Speed vs. Velocity • Speed is a scalar (how fast something is moving regardless of its direction). Ex: v = 20 mph • Speed is the magnitude of velocity. • Velocity is a combination of speed and direction. Ex: v = 20 mph at 15 south of west • The symbol for speed is v. • The symbol for velocity is type written in bold: v or hand written with an arrow: v

41. Speed vs. Velocity • During your 8 mi. trip, which took 15 min., your speedometer displays your instantaneous speed, which varies throughout the trip. • Your average speed is 32 mi/hr. • Your average velocity is 32 mi/hr in a SE direction. • At any point in time, your velocity vector points tangent to your path. • The faster you go, the longer your velocity vector.

42. m/s a = -3 = -3 m/s2 s Acceleration Acceleration – how fast you speed up, slow down, or change direction; it’s the rate at which velocity changes. Two examples: a = +2 mph/s

43. Velocity & Acceleration Sign Chart

44. Acceleration due to Gravity Near the surface of the Earth, all objects accelerate at the same rate (ignoring air resistance). This acceleration vector is the same on the way up, at the top, and on the way down! a = -g = -9.8 m/s2 9.8 m/s2 Interpretation: Velocity decreases by 9.8 m/s each second, meaning velocity is becoming less positive or more negative. Less positive means slowing down while going up. More negative means speeding up while going down.

45. vf = v0 + at • vavg = (v0 + vf)/2 • x = v0t + ½at2 • vf2 – v02 = 2ax Kinematics Formula Summary For 1-D motion with constant acceleration: (derivations to follow)

46. vavg = (v0 + vf)/2 will be proven when we do graphing. x = vt = ½ (v0 + vf)t = ½ (v0 + v0+at)t  x = v0 t+ at2 Kinematics Derivations a = v/t(by definition) a = (vf – v0)/t  vf = v0 + at (cont.)

47. vf= v0 +at  t = (vf – v0)/ax = v0t+at2  x =v0 [(vf – v0)/a] + a[(vf – v0)/a]2vf2 – v02 = 2ax Kinematics Derivations (cont.) Note that the top equation is solved for t and that expression for t is substituted twice (in red) into the x equation. You should work out the algebra to prove the final result on the last line.

48. Sample Problems • You’re riding a unicorn at 25 m/s and come to a uniform stop at a red light 20 m away. What’s your acceleration? • A brick is dropped from 100 m up. Find its impact velocity and air time. • An arrow is shot straight up from a pit 12 m below ground at 38 m/s. • Find its max height above ground. • At what times is it at ground level?

49. Multi-step Problems • How fast should you throw a kumquat straight down from 40 m up so that its impact speed would be the same as a mango’s dropped from 60 m? • A dune buggy accelerates uniformly at 1.5 m/s2 from rest to 22 m/s. Then the brakes are applied and it stops 2.5 s later. Find the total distance traveled. Answer: 19.8 m/s 188.83 m Answer:

50. x B A t C 1 – D Motion Graphing ! A … Starts at home (origin) and goes forward slowly B … Not moving (position remains constant as time progresses) C … Turns around and goes in the other direction quickly, passing up home