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8.2 Vector Addition

The golf problem shows how vectors can be added.  Vectors can also be used to show things such as velocity and force.  One of the main purposes for breaking things such as force into i, j, and k components is so we can find the result when different forces act on an object. When a number of forces act on an object and are broken down into i, j, and k vectors, the resulting force is simply the sum of these vectors.  The next section gives some examples of how to add velocity and force vectors. 

8.2.1  Velocity Vectors 

In the last problem, we added displacement vectors to track the motion of the golf ball to the hole. In the next problem, we’ll look at how velocity vectors combine to determine the direction of the shot.

On the next hole, Irene hits the ball with a velocity vector of 21i + 28j.  This time the units of the vector are in yards per second.  As the ball is traveling through the air, the wind is blowing with a velocity vector of 9i - 12j.  We can determine the final direction of the ball by adding the two vectors.

If we draw a picture of the two velocity vectors we can see how they add to the resultant vector.

You get the same resultant vector regardless of the order in which you add them.  The addition of vectors is therefore said to be commutative.

The subtraction of a vector is the same thing as adding a vector of equal magnitude in the opposite direction.

8.2.2 Force Vectors

Vectors are also used to find the result of several forces acting on an object.  Force vectors are added in the same way that velocity vectors are.  When vectors describe forces, the units are in pounds or the metric unit of force, the Newton (N).

Example 8.3:  During a running play in a football game, an offensive guard and tackle double-team a defensive guard, as shown in Figure 8.9, to create a hole for the running back.  As the block is made, all three players effectively move as one unit.  The players move as a result of the combination of forces that each applies. The play is viewed from above the field.  The force vectors of the players are as follows:

            defensive guard, DG:          -220j 

            offensive guard, OG:           5i + 200j 

            offensive tackle, OT:            -40i +140j

where the units of the vectors are in pounds.

(The forces are created by the players' legs.  The magnitude of the leg forces is like the leg force needed to move the same amount of weight in a squat lift.)

Determine the vector of the resultant force and its magnitude.

Solution:  As with the velocity vectors, the resultant of the addition of force vectors can be determined graphically by adding the force vectors head to tail. The three vectors are graphed as shown in Figure 8.9.

The resultant vector is determined by adding the three force vectors.

                        DG                                           -220j

                        OG                                    5i  + 200j

                        OT                                -40i  + 140j

                                                              -35i  + 120j

The resultant vector tells us that the net force of the blocks was a little to the left (-35i) but mostly forward (120j).   The magnitude of the resultant is:

The magnitude tells us how much the defensive guard is pushed. 

The defensive guard is pushed back and to the left by a force of 125 lb. by the two offensive linemen. 

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