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**Chapter 3 Motion in Two Dimensions**

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**Units of Chapter 3 Components of Motion**

Vector Addition and Subtraction Projectile Motion Relative Velocity

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3.1 Components of Motion An object in motion on a plane can be located using two numbers—the x and y coordinates of its position. Similarly, its velocity can be described using components along the x- and y-axes.

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**3.1 Components of Motion The velocity components are:**

The magnitude of the velocity vector is:

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3.1 Components of Motion The components of the displacement are then given by: Note that the x- and y-components are calculated separately.

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**3.1 Components of Motion The equations of motion are:**

When solving two-dimensional kinematics problems, each component is treated separately. The time is common to both.

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3.1 Components of Motion If the acceleration is not parallel to the velocity, the object will move in a curve:

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**3.2 Vector Addition and Subtraction**

Geometric methods of vector addition Triangle method:

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**3.2 Vector Addition and Subtraction**

The negative of a vector has the same magnitude but is opposite in direction to the original vector. Adding a negative vector is the same as subtracting a vector.

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**3.2 Vector Addition and Subtraction**

Vector Components and the Analytical Component Method If you know A and B, here is how to find C:

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**3.2 Vector Addition and Subtraction**

The components of C are given by: Equivalently,

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**3.2 Vector Addition and Subtraction**

Vectors can also be written using unit vectors:

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**3.2 Vector Addition and Subtraction**

Vectors can be resolved into components and the components added separately; then recombine to find the resultant.

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**3.2 Vector Addition and Subtraction**

This is done most easily if all vectors start at the origin.

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3.3 Projectile Motion An object projected horizontally has an initial velocity in the horizontal direction, and acceleration (due to gravity) in the vertical direction. The time it takes to reach the ground is the same as if it were simply dropped.

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3.3 Projectile Motion A projectile launched in an arbitrary direction may have initial velocity components in both the horizontal and vertical directions, but its acceleration is still downward.

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3.3 Projectile Motion The vertical motion is the same as if the object were thrown straight up or down with the same initial y velocity, and the horizontal velocity is constant.

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3.3 Projectile Motion The range of a projectile is maximum (if there is no air resistance) for a launch angle of 45°.

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3.3 Projectile Motion With air resistance, the range is shortened, and the maximum range occurs at an angle less than 45°.

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3.4 Relative Velocity Velocity may be measured in any inertial reference frame. At top, the velocities are measured relative to the ground; at bottom they are measured relative to the white car.

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3.4 Relative Velocity In two dimensions, the components of the velocity, and therefore the angle it makes with a coordinate axis, will change depending on the point of view.

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Review of Chapter 3 Two-dimensional motion is analyzed by considering each component separately. Time is the common factor.

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**Review of Chapter 3 Vector components:**

In projectile motion, the horizontal and vertical motions are determined separately.

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**Review of Chapter 3 Range is the maximum horizontal distance traveled.**

Relative velocity is expressed relative to a particular reference frame.

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