Vectors - Physics



Vectors

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Two-Dimensional Motion and Vectors

In the previous unit motion was limited to one-dimension in which objects could only move in two directions; either forward and backward or side to side. In order to describe these directions of motion mathematically, we used a positive or negative sign. In this unit, you will learn a new method to describe the motion of objects that do not travel in a straight line (two-dimensional).

Vector – a quantity that represents both magnitude and direction.

Give an example of a vector quantity for the following:

• Displacement – 5 meters East

• Velocity – 50. m/s South

• Acceleration – 4.7 m/s2 West

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|Graphic Representation of a Vector. |Elements of a Vector |

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|[pic] |Length represents magnitude. |

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| |Arrow represents direction. |

Measuring Vectors

Measure the magnitude and the direction of the vectors below using the given scales.

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Magnitude _~ 570_km_ Direction _at 53° N of E_ Magnitude _~ 89 m/s_ Direction _at 22° S of E_

Steps to Drawing a Vector

1. Determine scale.

2. Determine direction or angle.

3. Draw vector with appropriate magnitude (length).

Draw the following vectors:

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|A dog walks east for 20. meters. |

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|An airplanes flies with a velocity of 2400 km/hr, 15( north of west. |

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|A box is dragged to the right with a force of 30. Newtons at an angle of 20( above the horizon. |

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Using vectors represent the path of a person who walks 20. meters North, then 5.0 meters East.

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Component Vector – the horizontal and vertical parts that add up to form a vector.

Resultant Vector – the sum of one or more vectors.

Label the components and the resultant in your diagram above.

Adding Vectors

Two Methods for Adding Vectors

A tiger walks from its den 300. meters East, and then walks another 400. meters South. What is the displacement of the tiger from its den?

|Scale Method |Math Method |

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A boat travels north at 2.0 m/s across a river whose current is 3.0 m/s West. Determine the resultant velocity.

|Scale Method |Math Method |

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Adding Non-Perpendicular Vectors

Draw resultant from ____START____ to ____FINISH____.

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Adding Concurrent Vectors

Concurrent Vectors – two or more vectors originating from the same point.

Draw and label the resultant for the following concurrent vectors.

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More Concurrent Vectors

Draw the resultant for each vector.

| |Concurrent Vectors |Angle between vectors |

|1. |[pic] |0° |

|2. |[pic] |45° |

|3. |[pic] |90° |

|4. |[pic] |135° |

|5. |[pic] |180° |

What happened to the magnitude of the resultant as the angle between the two vectors increased?

The magnitude of the resultant decreased as the angle between the two vectors increased.

Equillibrium

Equillibrium – when the sum of all the vector acting on a single point is zero.

Example:

Equillibrant – a single vector that is exerted on an object to produce equilibrium, which is the same

magnitude as the resultant force but opposite in direction.

Draw the equilibrant for the following examples.

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Resolving Vectors into Components

Components – the horizontal and vertical parts that add up to form a vector.

Draw a horizontal and a vertical component making up the resultant below.

A man pushes a lawn mower at an angle of 30.0( from the horizon with a force of 100. Newtons. Draw a vector diagram showing the resultant force vector and its components.

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|Scale Method |Mathematical Method |

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|Scale: 1 cm = 100 N |Fy = F sin θ |

| |= (100. N) (sin 30.0°) |

| |= 50.0 cm |

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| |Fx = F cos θ |

| |= (100. N) (cos 30.0°) |

| |= 86.6 cm |

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The man pushing the mower begins to push it at a larger angle than in the problem before. What happens to the vertical and horizontal components of the force as the angle increases?

As the angle increases, the vertical component increases and the horizontal component decreases.

1. A horizontal force of 30. Newtons acts on a body concurrently with a 10. newtons force acting vertically (as shown below). Determine the resultant force.

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2. Professor Einstein walks 11.6 m, 55o north of east, as shown in the diagram. How many meters did he walk north? How many meters did he walk east?

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|Scale Method |Mathematical Method |

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|Measure x component and multiply by 2.0. |dy = d sin θ |

| |= (11.6 m) (sin 55°) |

|3.4 cm x 2.0 = 6.8 m East |= 9.5 m North |

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|Measure y component and multiply by 2.0. |dx = d cos θ |

| |= (11.6 m) (cos 55°) |

|4.7 cm x 2.0 = 9.4 m North |= 6.7 m East |

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Independence of Vectors

Describe what happens to the student’s speed when pushed at different angles.

|0° |45° |90° |135° |180° |

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|Speed |

|Increases |Increases |same |Decreases |decreases |

At which angle was the student’s speed influenced the least?

At 90° the student’s speed is least influenced.

A motorboat travels at 8.5 m/s, north straight across a river that has a current of 3.8 m/s east.

|Determine the boat's resultant velocity. | |

| |Velocity Diagram |

|v = (8.5 m/s)2 + (3.8 m/s)2 | |

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|= 9.3 m/s N of E | |

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|If the river is 100. m wide, how long it will take the boat to cross the river? | |

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|How far downstream will the boat be when it reaches the opposite shore? | |

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|d = vt = (3.8 m/s E) (12 s) |Displacement Diagram |

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|= 47 m E | |

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|How far will the boat actually travel? | |

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|110 m | |

Construct a statement regarding the independence of vectors.

Vectors perpendicular to one another are independent of each other.



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Scale: 1.0 cm = 15 m/s

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Scale: 1.0 cm = 200 km

2.85 cm

5.45 cm

Scale:

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Students will determine scale for the following practice problems.

How far did she walk?

25 meters

How far is she from where she started?

21 meters N of E

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üôîæüæÝüÎüÈü½ü½ü½ü½°½ü¦œüæü‘‰æüÈü½ü~üpProperties of Vectors

1. Vectors can be moved parallel to themselves.

2. Vectors can be added in any order.

3. Add / subtract vectors from head to tail.

4. Multiplying / dividing vectors with scalars results in vectors.

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LABEL RESULTANT

Ax = A cos θ

Ay = A sin θ

10. cm

5 cm

8.7 cm

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5.8 cm

4.7 cm

3.4 cm

v = (8.5 m/s)2 + (3.8 m/s)2

= 9.3 m/s N of E

3.8 m/s

8.5 m/s

46 m

100. m

Resultant

Equillibrant

Resultant

Resultant

Resultant

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