1Linear Kinematics of Human Movement (Ch 8)• Basic Kinematic QuantitiesPosition, Displacement, DistanceVelocity, SpeedAcceleration• Differentiation & Integration• Linear Kinematics of Locomotion• Uniform Acceleration / Projectile MotionCopyright © 2014Brian R. Umberger, Ph.D.University of Massachusetts AmherstPart IILinear Kinematics of Locomotion• In most forms of human locomotion, a basic pattern of movement is repeated over and over• In these situations, the magnitude of the velocity is equal to cycle length times cycle rate• For example:walking velocity = stride length  stride raterunning velocity = stride length  stride rateswimming velocity = stroke length  stroke ratewheelchair velocity = stroke length  stroke rateLinear Kinematics of RunningHow do stride length and stride rate contribute to forward progression over a wide range of running velocities?How can these results be explained?Linear Kinematics of SwimmingHow are the comparable patterns for swimming stroke length and rate similar to running stride length/rate, and how do they differ?Why are the results somewhat different for swimming?2Linear Kinematics of SprintingOver the last ~100 years, the world record time for the 100 m has improved considerablyworld record times for men’s 100 mLinear Kinematics of Sprinting• This means that sprinters’ average velocities have increased over time• Men’s 100 m1912 (Don Lippincott) v = 100 m / 10.6 s = 9.43 m/s2009 (Usain Bolt) v = 100 m / 9.58 s = 10.44 m/s11% improvement• Women’s 100 m1922 (Mary Lines) v = 100 m / 12.8 s = 7.81 m/s1988 (Flo-Jo) v = 100 m / 10.49s = 9.53 m/s22% improvementLinear Kinematics of Sprintinghttp://www.youtube.com/watch?v=8sKB8955n4U1988 Olympics100 m Men’s SprintBen Johnson*time 9.79 savg vel 10.21 m/speak vel 12.05 m/sCarl Lewistime 9.92 savg vel 10.08 m/speak vel 12.05 m/s*Stripped of gold medal and world record for use of the steroid stanozololLinear Kinematics of SprintingTwo questions for your consideration:What are the main characteristics of the velocity profile for a world class sprinter?How do they differ from the velocity profile for a washed-up former distance runner?3Linear Kinematics of SprintingFinally, how do the velocity and acceleration profiles for an elite sprinter compare to the theoretical optimum?data ---●---theory ______Theoretical optimal based on work by the Nobel laureate A.V. HillReview of VectorsOur previous examples were for motion in one coordinate direction; what if you have movement in more than one direction?Suppose that a hiker walked 5 km due east, then 3 km due north. What was her total displacement?To solve this problem we need to use vector composition (adding two or more vectors together to find the resultant vector)Vector CompositionGraphical Solution: Place the vectors tip-to-tail, and then draw in the resultant vector• Vectors along same line: just add or subtract magnitudes• Vectors not along same line: use the parallelogram law of vector additionThe composition of vectors with the same direction requires adding their magnitudesVector Composition4The composition of vectors with opposite directions requires subtracting their magnitudesVector CompositionVectors not along the same line require placing the vectors tip-to-tail to find the resultantVector CompositionVector AlgebraGraphical solutions are useful, but obtaining quantitative solutions requires the use of right triangle trigonometrysin  = A / C (opposite / hypotenuse)cos  = B / C (adjacent / hypotenuse)tan  = A / B (opposite / adjacent)C2= A2+ B2ABCVector CompositionNow, back to our walk: the hiker went 5 km due east, then 3 km due north What was her total displacement (R)?5 km3 kmRC2= A2+ B2R2= (5)2+ (3)2R = √(5)2+ (3)2R = 5.8 km NSWE5Vector CompositionThe displacement vector R had a magnitude of 5.8 kmWhat was the orientation of vector R?5 km3 kmRtan  = opp / adj = tan-1( opp / adj ) = tan-1( 3 / 5 )= 31.0NSWEMore on Vectors…Now, suppose the resultant vector is given, and the perpendicular components are requiredExample: a baseball is pitched with a velocity of 40 m/s at 10 deg above the horizontalHow fast is the ball traveling horizontally? How fast is the ball traveling vertically? = 10°V = 40 m/sVector Resolution is the process of replacing a single vector with two perpendicular vectors, such that the vector composition of the two perpendicular vectors yields the original vectorVector ResolutionRRRYRXRYRXBack To The Pitched BallWhat are the horizontal & vertical components of the resultant velocity vector of the pitched ball? = 10°V = 40 m/sVXVYsin  = opp / hyp = VY/ VVY= sin V VY= (sin 10) (40 m/s)VY= 6.9 m/scos  = adj / hyp = VX/ VVX= cos V VX= (cos 10) (40 m/s)VX= 39.4 m/s6Vectors: Another ExampleWhat is the world record in the long jump?What take off conditions are necessary to achieve this amazing displacement of the body?What dictates these relationships?8.95 m (29.4 ft)vh 9.5 m/svv 3.5 m/sEquations of Uniform AccelerationGalileo determined that objects fall to the Earth with constant acceleration, from which we get the following three equations:vf= vi+ ats = vit + ½at2vf2= vi2+ 2asThe most common case ofuniform acceleration relevantto biomechanics is the motionof projectilesGalileo Galilei (1564-1642)Projectile Motion• A projectile is an object that is in free fall and is subject only to the acceleration due to gravity (and possibly air resistance)• In many cases (long jump, high jump, shot put, free throw), air resistance is negligible (i.e., we can safely ignore it)• In other cases (skydiver, flight of a golf ball, a tennis serve), acceleration resulting from air resistance needs to be account forProjectile Motion• If air resistance is negligible, projectile motion is simply a special case of uniform acceleration • With no air resistance, the path followed by a projectile will be a parabola7Projectile Motion• Even with complex body segment motions, the center of mass will follow a parabolic path when the body is airborneProjectile Motion• During running, your foot is on the ground only 20-30% of the time; the rest of the time the body is a projectile• Thus, gravity greatly influences the way we runFrom: E. Muybridge (1830-1904)Projectile MotionHorizontal and vertical components of velocity are independent, and