1. How to improve running using Math! Camille Olson MAT 170 MWF 10:45

2. The Study of Motion Scalar Quantities: a quantity that is fully described by its magnitude Scalar Quantities: a quantity that is fully described by its magnitude Ex. The box weighs 200 grams. Ex. The box weighs 200 grams. Vector Quantities: a quantity fully described by its magnitude and direction. Vector Quantities: a quantity fully described by its magnitude and direction. Ex. The church is 20 miles North East of the school. Ex. The church is 20 miles North East of the school.

3. The Study of Motion Vector Quantities Vector Quantities Force Force Displacement Displacement Acceleration Acceleration Velocity Velocity Scalar Quantities Scalar Quantities Distance Distance Mass Mass Energy Energy

4. Vectors: Force Force is that which can cause an object with mass to accelerate Force is that which can cause an object with mass to accelerate massaccelerate massaccelerate Newton’s Second Law: an object with constant mass will accelerate in proportion to the net force acting upon it and in inverse proportion to its mass. Newton’s Second Law: an object with constant mass will accelerate in proportion to the net force acting upon it and in inverse proportion to its mass.net forcenet force Forces acting on three-dimensional objects may also cause them to rotate or deform, or result in a change in pressure. Forces acting on three-dimensional objects may also cause them to rotate or deform, or result in a change in pressure.rotatedeformpressurerotatedeformpressure Torque: The tendency of a force to cause angular acceleration about an axis Torque: The tendency of a force to cause angular acceleration about an axisangular accelerationangular acceleration

5. There are Three External Forces that act on the body at all times: There are Three External Forces that act on the body at all times: Gravity Gravity Ground Reaction Ground Reaction Momentum Momentum our posture and the actions we take with our bodies (or our movement patterns), and how we apply these to our running, will determine if these three forces work for us or against us. our posture and the actions we take with our bodies (or our movement patterns), and how we apply these to our running, will determine if these three forces work for us or against us. Forces on the Body When Running

6. Utilizing the Three Forces Learning about these forces improves the way we direct ourselves and put ourselves into our running form. Learning about these forces improves the way we direct ourselves and put ourselves into our running form. It maximizes efficiency to learn about these forces, how they operate on us, and what we can do to work in harmony with them. It maximizes efficiency to learn about these forces, how they operate on us, and what we can do to work in harmony with them. It’s like driving a car with the hand brakes on It’s like driving a car with the hand brakes on We get used to them being on and we fail to notice the drag it creates on our bodies. We get used to them being on and we fail to notice the drag it creates on our bodies.

7. Gravity Main concern: Good bodily alignment or what is commonly known as good posture. Main concern: Good bodily alignment or what is commonly known as good posture. A well aligned and relaxed body, and getting a mild forward lean into the form to establish a gentle falling action, the runner can take advantage of the gravity factor, with a continuous falling and catching effect. A well aligned and relaxed body, and getting a mild forward lean into the form to establish a gentle falling action, the runner can take advantage of the gravity factor, with a continuous falling and catching effect. Any break in the postural alignment will diminish the efficiency by either causing a disruption in the energy flows of the body itself or by inhibiting the freedom of movement at the joints. Any break in the postural alignment will diminish the efficiency by either causing a disruption in the energy flows of the body itself or by inhibiting the freedom of movement at the joints. This generally results in over striding, breaking and unnecessary pounding. This generally results in over striding, breaking and unnecessary pounding.

8. Ground Reaction Newton: “For every action, there is an opposite and equal reaction.” Newton: “For every action, there is an opposite and equal reaction.” For ground reaction to be beneficial, certain physical principles must be honored: For ground reaction to be beneficial, certain physical principles must be honored: Muscles must be resilient Muscles must be resilient the right measure of contraction and relaxation must be present the right measure of contraction and relaxation must be present There needs to be proper use and balance in the muscular system. There needs to be proper use and balance in the muscular system. Failure to achieve this will result in a variety of foot, ankle, leg, and knee injuries with increases in mileage and age. Failure to achieve this will result in a variety of foot, ankle, leg, and knee injuries with increases in mileage and age.

9. Momentum (the most important!) A body set in motion will remain in motion until a retarding force acts on it. A body set in motion will remain in motion until a retarding force acts on it. Opposing forces: friction, wind, gravity, variations in terrain. Opposing forces: friction, wind, gravity, variations in terrain. Biggest refrain = overly tense muscles, poor postural alignment, biomechanical form. Biggest refrain = overly tense muscles, poor postural alignment, biomechanical form. Being centered in the body and moving as one unit, all parts, muscles, and body segments contributing to the effort in an integrated and harmonious way helps develop and maintain momentum in running. Being centered in the body and moving as one unit, all parts, muscles, and body segments contributing to the effort in an integrated and harmonious way helps develop and maintain momentum in running.

10. Math of Momentum Momentum = product of mass and velocity Momentum = product of mass and velocity P = m x v P = m x v A conserved quantity, meaning that the total momentum of any closed system cannot change. A conserved quantity, meaning that the total momentum of any closed system cannot change.conservedclosed systemconservedclosed system SI unit kg·m/s, or, equivalently, N·s SI unit kg·m/s, or, equivalently, N·s SIkgm/sNs SIkgm/sNs

11. The Law of the Pendulum "The speed at which a pendulum swings depends on the length of the pendulum, not on the amount of weight at the bottom.“ "The speed at which a pendulum swings depends on the length of the pendulum, not on the amount of weight at the bottom.“ When you want to get a pendulum to swing faster you need to shorten it. When you want to get a pendulum to swing faster you need to shorten it. A pendulum is an excellent example of the conservation of momentum. A pendulum is an excellent example of the conservation of momentum. So…how does this apply to running? So…how does this apply to running?

12. Running like a Pendulum! Arms and legs are both pendulums Arms and legs are both pendulums When you swing your arm, your shoulder is the top of a pendulum and your hand is at the bottom. When you swing your arm, your shoulder is the top of a pendulum and your hand is at the bottom. Likewise, when you swing your leg, your hip is at the top of a pendulum and your foot is at the bottom. Likewise, when you swing your leg, your hip is at the top of a pendulum and your foot is at the bottom. Therefore, in order to get your pendulum (either your arms or your legs) to swing faster, you need to shorten them. Therefore, in order to get your pendulum (either your arms or your legs) to swing faster, you need to shorten them. Bending either the arms or the legs shortens the pendulum because you're moving the center of gravity towards the top of the pendulum. Bending either the arms or the legs shortens the pendulum because you're moving the center of gravity towards the top of the pendulum. When you swing faster, you move faster. When you swing faster, you move faster.

13. Example Using Vectors for Running ex. I ran 2 miles north and 5 miles east this morning. What is the distance of the quickest route back to my original location? a = a = length of a = = sqroot (25 + 4) = 5.39 miles