Vectors and Motion in Two Dimensions

AP Topic 1.5a

Denver Art Museum, Studio Libeskind Architects, 2006

Learning Objective

Describe the perpendicular components of a vector.

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Relevant equations:

Vector Addition

• Vector quantities can be added together using what is called the “head-to-tail” method.
• The tail of one vector is placed at the head of another vector.
• The resulting vector from the tail of the first vector to the head of the second vector is called the “resultant”.

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Click here to play with a simulation.

+

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Resultant Vectors

Vectors can be mathematically modeled as the resultant of two perpendicular components.

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x-component

y-component

As you just learned about vector addition, the vector in question (green) is the sum of the two component vectors.

Vector Resolution

Vectors can be resolved into components using a chosen coordinate system.

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You can see that the horizontal component is +3 units long and the vertical component is +6 units long.

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These components add up to the 6.7 unit original vector.

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6.7 units

3 units

6 units

Vector Resolution

How can we find the angle of this vector (from the +x axis)?

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6.7 units

3 units

6 units

θ = 63.6^o above the +x axis

θ

Vector Resolution

Here’s another one.

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You can see that the horizontal component is -5 units long and the vertical component is -7 units long.

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These components add up to the 8.6 unit original vector.

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8.6 units

Vector Resolution

Calculate the angle from the –x axis.

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8.6 units

θ = 54.5^o below the –x axis

θ

Vector Resolution

Vectors can be resolved into perpendicular components using trigonometric functions and relationships.

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Let’s mathematically resolve this vector and then confirm the results on the coordinate system.

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57.5^o

6.5 units

Vector Resolution

57.5^o

6.5 units

Vector Resolution

57.5^o

6.5 units

Practice 1

Resolve the following vector into it’s horizontal and vertical components:

A cannon fires a cannon ball with velocity of 92.0 m/s at an angle of 26.5^o above the ground.

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Draw a picture!

Vector Resolution

26.5^o

92.0 m/s

Practice 2

Resolve the following vector into it’s horizontal and vertical components:

Takeet Easy pulls a box to the right with a rope with a force of 53 N. If the rope is at an angle of 42^o from the ground, what are the components of the force?

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Draw a picture!

Vector Resolution

42^o

53 N