ESS 303 1st EditionFinal Exam Study GuideLinear Kinematics  Linear: in a straight line (from point A to point B) Angular: rotational (from angle A to angle B) Kinematics VS Kinetics Kinematics: description of motion without regard for underlying forces Acceleration Velocity Position Kinetics: determination of the underlying causes of motion (i.e., forces) Linear Kinematics The branch of biomechanics that deals with the description of the linear spatial and temporal components of motion Describes transitional motion (from point A to point B) Uses reference systems 2D: X & Y axis 3D: X, Y & Z axis Position: location in space relative to a reference Scalars and vectors Scalar quantities: described fully by magnitude (mass, distance, volume, etc) Vectors: magnitude and direction (the position of an arrow indicates direction and the length indicates magnitude) Distance: the linear measurement of space between points Displacement: area over which motion occurred, straight line between a starting and ending point Speed: distance per unit time (distance/time) Velocity: displacement per unit time or change in position divided by change in time (displacement/time) Graph Basics SI Units Systeme International d’Units Standard units used in science Typically metric Mass: Kilograms Distance: Meters Time: Seconds Temperature: Celsius or kalvin More Terms Acceleration: change in velocity divided by change in time (Δ V / Δ t) (m/s)/s Acceleration of gravity: 9.81m/s2 Differentiation: the mathematical process of calculating complex results from simple data (e.g., using velocity and time to calculate acceleration) Derivative: the solution from differentiation Integration: the opposite of differentiation (e.g., calculation of distance from velocity and time) Speed = d / t Velocity = Δ position / Δ t Acceleration = Δ V / Δ t Slope = rise / run Resultant = √(X2 + Y2) Remember: A2 + B2 = C2 SOH CAH TOA Sin θ = Y component / hypotenuse Cos θ = X component / hypotenuse Tan θ = Y component / X component  Sample Problems A swimmer completes 4 lengths of a 50m pool What distance was traveled? What was the swimmer’s displacement? Move from point (3,5) to point (6,8) on a graph What was the horizontal displacement? What was the vertical displacement? What was the resultant displacement? Sample Problems A runner accelerates from 0m/s to 4.7m/s in 3.2 seconds What was the runner’s rate of acceleration? Someone kicks a football so that it travels at a velocity of 29.7m/s at an angle of 22° above the ground What was the vertical component of velocity? What was the horizontal component of velocity?Angular Kinematics The branch of biomechanics that deals with the description of the angular components of motion Uses degrees or radians to describe position and/or movement Degree: 360° in a circle Radian: the length of 1 radius along the arc of a circle 1 radian = 57.3 degrees Angular Kinematics In the drawing to the right – A, B & C have the same angular displacement or rotation A, B & C have different linear displacements Angular Kinematics θ = S/R θ = angle in radians S = displacement along the arc R = radius If radius A = 1m, radius B = 2m, radius C = 3m and each had a rotation of 90°, what were the displacements of each? Angular Kinematics 90° = 1.57 radians SA = 1.57rad * 1m SA = 1.57m SB = 1.57rad * 2m SB = 3.14m SC = 1.57rad * 3m SC = 4.71m Angle Types Relative: angle between segments Absolute: describes the orientation of an object in space Right Hand Rule Today’s Formulas 1 radian = 57.3 degrees  θ = S/R (remember to use radians here) Tan θ = (Yproximal – Ydistal)/(Xproximal – Xdistal) Angular speed = angular distance/time Angular velocity (ω) = ∆θ / ∆t Angular acceleration (α) = ∆ω / ∆t Problems A figure skater turns 6 ½ times What was the angular distance traveled? What was the angular displacement? While watching a golf swing, you note that the angular velocity at time1 (0.05s) was 6.5rad/s and at time2 (0.54s) was 15.87rad/s What was the angular acceleration?Projectiles Which Will Hit First? Put Things Together (4 Steps) Step #1: Calculate the X and Y components of movement. VX0 = V0 * Cos θ VY0 = V0 * Sin θ Step #2: Calculate the maximum height. How far up did it go? Yup = (VY2 – VY02)/2a Ydown: How far down will it fall (think about it)? Put Things Together (4 Steps) Step #3: Calculate the hang-time Y = ½at2 t = √(2Y)/a Remember to add time up and time down Step #4: Calculate the range (the horizontal distance) X = VX0 * t Problems You drop a penny from the top of a 2000 meter-high building How long will it take to hit the ground? How fast will it be going when it hits? Problems A player kicked a football giving it a velocity of 20m/s at an angle of 37°. It was caught byanother player at a height of 1.5meters. What were the X and Y components of velocity? How high did it go? How long was it in the air? How far apart were the players? Quiz 2 Notes Next week Bring a calculator (with trig. functions) Formula sheet provided (same as one on the web) Scratch paper is recommended About 16 total points About 5 problems ranging in points from 2 to 7 Show your work and include units Quiz 2 Sample Problems A runner completes 17 laps of a quarter-mile track. What was his or her distance and displacement? Distance = 4.25 Miles Displacement = 0 Miles Move on a grid from point (3,6) to point (8,10). What are your horizontal, vertical, and resultant displacements? Horizontal Displacement = 5 Vertical Displacement = 4 Resultant Displacement = 6.40 More Quiz 2 Sample Problems A ballerina twirls 4½ times. What was her angular distance in degrees and radians? Angular Distance = 1620 Degrees Angular Distance = 28.27 Radians While watching a golf swing, you note that the angular velocity at the .18 second mark was 2.5 radians/second. You also see that the angular velocity at the .41 second mark was 12.02 radians/second. What was the angular acceleration?  = (/t)   = (12.02-2.5)/.41-.18) = (9.52/0.23) = 41.39