Chapter 3

Kinematics in Two Dimensions; Vectors

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3-1 Vectors and Scalars

A vector has magnitude as well as direction. (displacement, velocity, force, momentum)

A scalar has only a magnitude. (mass, time, temperature)

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3-2 Addition of Vectors – Graphical Methods

For vectors in one dimension, simple addition and subtraction are all that is needed.

You do need to be careful about the signs, as the figure indicates.

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Addition of vectors that are vertical or horizontal only

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If the motion is in two dimensions, the situation gets more complicated.

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2 Dimensional Kinematics• We need to use

vector diagrams to describe the motion

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Describing the direction or angle

of the vector7

Describing the magnitude of the vector

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3-2 Addition of Vectors – Graphical Methods

Even if the vectors are not at right angles, they can be added graphically using the “tail-to-tip” method.

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3-2 Addition of Vectors – Graphical MethodsIf the paths are at right angles to one another; we can find the resultant by using the Pythagorean Theorem.

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Using the Pythagorean Theorem to find the magnitude of the vector

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3-4 Adding Vectors by Components

If the components are perpendicular, they can be found using trigonometric functions.

Remember: SOH-CAH-TOA! 13

To determine the direction or angle of the resultant vector

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Tan 111

Tan 1)

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Vertical vector components are noted with a y and horizontal with an x

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Fx

Fy

Fx

Fx

Fx

Fy

Fy

Fy

3-4 Adding Vectors by Components

Adding vectors:1. Draw a diagram.

2. Choose x and y axes.

3. Resolve each vector into x and y components.

4. Calculate each component using trig functions.

5. Add the components in each direction.

6. To find the magnitude and direction of the vector, use:

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3-5 Projectile MotionA projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola.

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Can be understood by analyzing the horizontal and vertical motions separately. The x and y components are independent of each other!

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HorizontalMotion (x)

VerticalMotion (y)

Forces(Present? - Yes or No. If present, what direction?)

NoYes The force of gravity acts downward

Acceleration(Present? - Yes or No. If present, what direction?)

No Yes "g" is downward at 9.8 m/s/s

Velocity(Constant or Changing?)

Constant Changing (by 9.8 m/s each second)

Analyzing parabolic motion

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At 1 sec time intervals

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3-5 Projectile MotionIf an object is launched at an initial angle of θ0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component.

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Objects launched at an angle

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Our previously learned equations still work! We just have to analyze the x and y components of the

motion separately.

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3-6 Solving Problems Involving Projectile Motion

1. Read the problem carefully, and choose the object(s) you are going to analyze.

2. Draw a diagram.

3. Choose an origin and a coordinate system.

4. Decide on the time interval; this is the same in both directions, and includes only the time the object is moving with constant acceleration g.

5. Examine the x and y motions separately.

6. List known and unknown quantities. Remember that vx never changes, and that vy = 0 at the highest point.

7. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them.

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It’s all relative!

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Magnitude of the resultantA2 + B2 = R2

(100 km/hr)2 + (25 km/hr)2 = R2 R= 103.1 km/hr

Direction of resultanttan Ө= opp/adjtan Ө = (25/100)Ө = 14.0 degrees

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Each velocity is labeled first with the object, and second with the reference frame in which it has this velocity. Therefore, vWS is the velocity of the water in the shore frame, vBS is the velocity of the boat in the shore frame, and vBW is the velocity of the boat in the water frame.

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3-8 Relative VelocityIn this case, the relationship between the three velocities is:

(3-6)

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References

Giancoli, Douglas. Physics: Principles with Applications 6th

Edition. 2009. http://www.physicsclassroom.com

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