lesson 9 linear kinetics impulse momentum applications

8/3/2019 Lesson 9 Linear Kinetics Impulse Momentum Applications

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Lesson 9: Linear Kinetics: Linear Impulse Momentum Applications

Assignment

Read Chapter 5.

Complete the review questions.

Take the quiz for Lesson 9.

Objectives

Define horizontal impulse-momentum. Define vertical impulse-momentum. Define braking and propulsion impulse. Solve impulse-momentum problems.

Impulse Momentum Relationship

Impulse can be computed either by measuring the force and the time of force application, or by

measuring the change in velocity before and after the force is applied and computing the change in

momentum.

Average Force times Time = Change in Linear Momentum

The Area under the Force Time Curve = Change in Linear Momentum

The two equations above are general equations for computing impulse. They do not differentiate

between horizontal and vertical impulse. In this lesson we will differentiate between horizontal and

vertical impulse and then apply the above equations to data sets on running, walking and the jumping.

Horizontal Force and Horizontal Impulse in Running

Braking Impulse

The figure below shows horizontal force and horizontal velocity for a runner during ground contact. In

addition, the free-body diagram is shown for three phases of the motion: heel-strike, mid-stance and

toe-off. Begin by examining the horizontal force from t = 0 to t = .112 s, notice that the force is negative.

In the FBD for heel-strike the horizontal force (Rx) is shown as negative. Since this force is opposing the

runners motion it will cause the runner to decelerate, also notice that the Rx force causes the runners

center of mass to have a negative horizontal acceleration (ax). In biomechanics, it is customary to refer

to the impulse caused by this force which opposes motion as a braking impulse. The result of this

braking impulse is to decrease the horizontal velocity of the runners center of mass from Vx = 3.0 m/s

at heel-strike to Vx = 2.79 m/s at mid-stance. A propulsion impulse is defined as an impulse where the

direction of the impulse is the same as the direction of motion. In the horizontal force graph below the


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force Rx is positive from t = .113 to t = .235 s, the propulsion impulse from this positive force causes the

runners horizontal velocity to increase from mid-stance to toe-off.

Example: Use the average force and area under the curve impulse-momentum formulas to compute the

Vxffor the runner below.



After experiencing the braking impulse the runners center of mass has slowed down to 2.79 m/s. Next

we will now compute the impulse for the propulsion phase.

Propulsion Impulse

The propulsion phase begins at mid-stance at t = .113 s. At the start of propulsion the horizontal

velocity is 2.79 m/s, so the final velocity from the previous step (braking phase) becomes the initial

velocity for propulsion.



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{IM-App-0.gif}


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Vertical Force and Vertical Impulse in Running

When computing vertical impulse it is necessary to account for the vertical impulse caused by the force

of gravity. The figure below shows the free-body diagram for a runner. Substituting the kinematicformula for vertical velocity into Newtons second law of motion gives the vertical impulse-momentum

equation. Similar to the general equation for impulse-momentum, vertical impulse-momentum can be

computed using either the average vertical force or integrating to get the area under the vertical force-

time curve.

{IM-App-1.gif}

The figure below shows vertical force and vertical velocity for a runner during ground contact. In


toe-off. The vertical force (Ry) is positive throughout ground contact. At heel-strike the vertical velocity

of the center of mass (Vy) is negative. Since the force Ry is opposing the runners motion it will cause

the vertical velocity of the runners center of mass to change to change from negative (Vyi= .6 m/s) atheel-strike to positive (Vyf= .48 m/s) at toe-off.

Example: Use the average force and area under the curve impulse-momentum formulas to compute the

Vyffor the runner below.




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{IM-App-2.gif}

Horizontal Force and Horizontal Impulse in Walking

The horizontal force and horizontal velocity for walking are shown in the figure below. Horizontal force

in walking typically begins with a brief propulsion phase, following by braking and propulsion.

Example: Use the average force and formula to compute the horizontal velocity (Vx) for each phase of

the horizontal force curve for the walking curve below.


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First Propulsion Phase

Braking Phase

Second Propulsion Phase


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{IM-App-3.gif}

Vertical Force and Vertical Impulse in Walking

The figure below shows vertical force and vertical velocity for a walker during ground contact. In


toe-off. The vertical force (Ry) is positive throughout ground contact. At heel-strike the vertical velocity

of the center of mass (Vy) is negative. Since the force Ry is opposing the walkers motion it will cause

the vertical velocity of the walkers center of mass to change to change from negative (Vy i= .18 m/s) at

heel-strike to positive (Vyf= .13 m/s) at toe-off.

Example: Use the area under the curve impulse-momentum formula to compute the Vyffor the walker

below.



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()

{IM-App-4.gif}

Impulse Momentum in the Vertical Jump

The vertical force and vertical velocity for a vertical jump are shown in the figure below. At the

start of the jump the vertical velocity Vy = 0 m/s. At approximately 0.26 s the vertical velocity

attains the greatest downward value of 0.8 m/s. The jumper stops moving downward at t = .42

s and finally at takeoff the vertical velocity is 2.06 m/s. Use the area under the force time

formula to compute the vertical velocity at takeoff as follows.


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The red dashed line and red circles on the graph below shows the point in the motion when the jumper

is moving downward the fastest, at this point the jumpers center of mass has a vertical velocity of 0.8

m/s and the vertical force is equal to the jumpers body weight. The blue dashed line and blue circles on

the velocity and force curves indicate the point in the motion when the jumper has stopped moving

down and is about to begin moving upward. From the perspective of muscle mechanics this point is

then end of the eccentric loading of the knee extensors and the hip extensors.

{IM-App-5.gif}


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Relationship between Force and Acceleration

Remember from Newtons second law of motion that acceleration is directly related to force and that

acceleration occurs in the direction of the net force. The shape of the acceleration curve is ALWAYS the

same as the shape of the force curve. If you divide the force curve by mass then you have an

acceleration curve. For a vertical force you must subtract the weight of the object/person before

dividing by mass. The graph below shows the direct relationship between force and acceleration for a

vertical jump.

{IM-App-5a.gif}

Speed Up Slowing Down and Constant Velocity Run/Walk Curves

The graph below shows typical horizontal force curves with braking and propulsion impulses and the

change in horizontal velocity for: speeding up, constant velocity and slowing down. When running or

walking at a constant velocity the area of braking is equal to the area of propulsion. According to the

impulse momentum relationship the change in horizontal velocity will be zero or close to zero. When

speeding up (accelerating) during walking or running the area of propulsion will be greater than the area

of braking and the change in velocity will be a positive number. Finally when slowing down

(decelerating) the area of braking will be greater than the area of propulsion and the change in

horizontal velocity will be negative.


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{IM-App-6.gif}

Comparison between Horizontal and Vertical Force for Walking and Running

The figure below shows both horizontal and horizontal forces for walking and running. The magnitude

of the forces for running is greater than the magnitude for walking. The time of ground contact is much

longer for walking than running.


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{IM-App-7.gif}

Computing Acceleration and Velocity in Excel from a Force Curve

Download the Excel File: Lesson 9 walk Practice.xls

When you first open up the file it should look like the figure below. Follow the instructions below to

compute ax, ay, vx, vy and impulses.

{IM-Excel-0-a.png}

Horizontal Acceleration

To compute ax enter the following equation in cell D2 and copy it down the rest of the column to row

757, where the data end.


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=B2/$L$2

The equation above is derived as follows.

{IM-App-8.gif}

Vertical Acceleration

To compute ax enter the following equation in cell E2 and copy it down the rest of the column to row


=(C2+($L$2*-9.8))/$L$2


{IM-App-9.gif}

Horizontal Velocity

To compute ax enter the following equation in cell F3 and copy it down the rest of the column to row


=((B3*0.001)+($L$2*F2))/$L$2


{IM-App-10.gif}


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Vertical Velocity

To compute ax enter the following equation in cell G3 and copy it down the rest of the column to row


= (((C3 + ($L$2*-9.8))*0.001) + ($L$2*G2)) / $L$2


{IM-App-11.gif}

Computing Impulses in Excel

The figure below shows the horizontal force curve and the Horizontal Impulse-Momentum section of the

Excel spreadsheet. After making a graph of horizontal force, you must enter the start and end row

numbers where the force curve CROSSES the Y = 0 AXIS. These row numbers should be placed in

columns M and N. The graph below show the corresponding rows where the force curve crosses the Y =

0 axis.

{IM-Excel-0.png}

To find these row numbers look in Column B. As shown in the figure below the force is negative from

rows 2 8 and row 9 is the start of the positive phase. Enter 2 in cell M6 and 8 in cell N6 to indicate the


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start and end of the first negative phase. Then scroll down to find the end of the positive phase that

begins in row 9.

{IM-Excel-1.png}

Enter 9 in cell M7 and 51 in cell N7 to indicate the start and end of the first positive phase. Then scroll

down to find the end of the negative phase that begins in row 52.

{IM-Excel-2.png}

Enter 52 in cell M8 and 455 in cell N8 to indicate the start and end of the first negative phase. Then

scroll down to find the end of the positive phase that begins in row 456.

{IM-Excel-3.png}


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Finally, enter 456 in cell M9 and 757 in cell N9 to indicate the start and end of the first negative phase.

Then scroll down to find the end of the positive phase that begins in row 9.

{IM-Excel-4.png}

When you have completed your work you can compare the results to the completed file.

Lesson 9 walk.xls


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The horizontal and vertical impulse-momentum variable sections are designed to automatically compute

impulses using both the change in momentum and the average force times dt. Once you enter the start

and end rows, it computes the velocity for the start and endpoints, the time between the two points,

the average force and the impulse using both methods. The horizontal and vertical impulse sections are

programmed to compute the impulse variables for up to 8 different regions on a curve.

{IM-Excel-5.png}

lesson 9 linear kinetics impulse momentum applications

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