# UMass Amherst KIN 430 - linear-kinematics-2 (14 pages)

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- School:
- University of Massachusetts Amherst
- Course:
- Kin 430 - Biomechanics

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Linear Kinematics of Human Movement Ch 8 Linear Kinematics of Locomotion Part II Basic Kinematic Quantities Position Displacement Distance Velocity Speed Acceleration Differentiation Integration Linear Kinematics of Locomotion Uniform Acceleration Projectile Motion 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 rate running velocity stride length stride rate swimming velocity stroke length stroke rate wheelchair velocity stroke length stroke rate Copyright 2014 Brian R Umberger Ph D University of Massachusetts Amherst Linear Kinematics of Running How 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 Swimming How 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 1 Linear Kinematics of Sprinting Over the last 100 years the world record time for the 100 m has improved considerably Linear Kinematics of Sprinting This means that sprinters average velocities have increased over time Men s 100 m 1912 Don Lippincott v 100 m 10 6 s 9 43 m s 2009 Usain Bolt v 100 m 9 58 s 10 44 m s 11 improvement Women s 100 m 1922 Mary Lines v 100 m 12 8 s 7 81 m s 1988 Flo Jo v 100 m 10 49s 9 53 m s 22 improvement world record times for men s 100 m Linear Kinematics of Sprinting Linear Kinematics of Sprinting 1988 Olympics 100 m Men s Sprint Two questions for your consideration Ben Johnson What are the main characteristics of the velocity profile for a world class sprinter time 9 79 s avg vel 10 21 m s peak vel 12 05 m s Carl Lewis time 9 92 s avg vel 10 08 m s peak vel 12 05 m s Stripped of gold medal and world record for use of the steroid stanozolol How do they differ from the velocity profile for a washedup former distance runner http www youtube com watch v 8sKB8955n4U 2 Linear Kinematics of Sprinting Review of Vectors Finally how do the velocity and acceleration profiles for an elite sprinter compare to the theoretical optimum Our 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 Theoretical optimal based on work by the Nobel laureate A V Hill data theory Vector Composition Graphical Solution Place the vectors tip totail and then draw in the resultant vector Vector Composition The composition of vectors with the same direction requires adding their magnitudes Vectors along same line just add or subtract magnitudes Vectors not along same line use the parallelogram law of vector addition 3 Vector Composition Vector Composition The composition of vectors with opposite directions requires subtracting their magnitudes Vectors not along the same line require placing the vectors tip to tail to find the resultant Vector Algebra Vector Composition Now back to our walk the hiker went 5 km due east then 3 km due north Graphical solutions are useful but obtaining quantitative solutions requires the use of right triangle trigonometry What was her total displacement R N sin A C opposite hypotenuse cos B C adjacent hypotenuse tan A B opposite adjacent W C2 A2 B2 C A R2 5 2 3 2 C2 A2 B2 B R 5 2 3 2 R 5 8 km E S R 3 km 5 km 4 Vector Composition More on Vectors Now suppose the resultant vector is given and the perpendicular components are required The displacement vector R had a magnitude of 5 8 km Example a baseball is pitched with a velocity of 40 m s at 10 deg above the horizontal What was the orientation of vector R N W tan opp adj tan 1 opp adj tan 1 V 40 m s 3 km 31 0 How fast is the ball traveling horizontally How fast is the ball traveling vertically S R 3 5 E 10 5 km Vector Resolution Back To The Pitched Ball Vector 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 vector What are the horizontal vertical components of the resultant velocity vector of the pitched ball V 40 m s 10 R RY R RX RY RX VX VY sin opp hyp VY V cos adj hyp VX V VY sin V VX cos V VY sin 10 40 m s VX cos 10 40 m s VY 6 9 m s VX 39 4 m s 5 Vectors Another Example Equations of Uniform Acceleration What is the world record in the long jump Galileo determined that objects fall to the Earth with constant acceleration from which we get the following three equations What take off conditions are necessary to achieve this amazing displacement of the body What dictates these relationships vv 3 5 m s vh 9 5 m s vf vi at s vit at2 vf2 vi2 2as The most common case of uniform acceleration relevant to biomechanics is the motion of projectiles Galileo Galilei 1564 1642 8 95 m 29 4 ft Projectile Motion 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 If air resistance is negligible projectile motion is simply a special case of uniform acceleration In many cases long jump high jump shot put free throw air resistance is negligible i e we can safely ignore it With no air resistance the path followed by a projectile will be a parabola In other cases skydiver flight of a golf ball a tennis serve acceleration resulting from air resistance needs to be account for 6 Projectile Motion Projectile Motion Even with complex body segment motions the center of mass will follow a parabolic path when the body is airborne 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 run From E Muybridge 1830 1904 Projectile Motion Horizontal and vertical components of velocity are independent and can be treated separately Gravity will cause the vertical component of the velocity vector to change during the flight Projectile Motion The horizontal component of the velocity vector will be constant for the whole flight vertical velocity is zero right here 7 Projectile Motion Projectile Trajectory Thus if air resistance is negligible To influence the trajectory of a projectile

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