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Vectors & Scalars Physics 11. Vectors & Scalars. A vector has magnitude as well as direction. Examples: displacement, velocity, acceleration, force, momentum A scalar has only magnitude Examples: time, mass, temperature. Vector Addition – One Dimension. - PowerPoint PPT Presentation
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Vectors & Scalars•A vector has magnitude as well as direction.
•Examples: displacement, velocity, acceleration, force, momentum
•A scalar has only magnitude
•Examples: time, mass, temperature
Vector Addition – One Dimension
A person walks 8 km East and then 6 km East.
Displacement = 14 km East
A person walks 8 km East and then 6 km West.
Displacement = 2 km
Vector Addition
21 DDDR
22
21 DDDR
Example 1: A person walks 10 km East and 5.0 km North
kmkmkmDR 2.11)0.5()0.10( 22
RD
D2sin
0121 5.26)2.11
0.5(sin)(sin
km
km
D
D
R
Order doesn’t matter
Graphical Method of Vector Addition Parallelogram Method
Helpful hints about parallelograms:•All four angles add to equal 360o
•Opposite angles are equal
Subtraction of Vectors
• Negative of vector has same magnitude but points in the opposite direction.
• For subtraction, we add the negative vector.
Multiplication by a Scalar
• A vector V can be multiplied by a scalar c• The result is a vector cV that has the same direction but a
magnitude cV• If c is negative, the resultant vector points in the opposite
direction.
Adding Vectors by Components• Any vector can be expressed as the sum of two other
vectors, which are called its components (i.e. Vx & Vy).
• Components are chosen so that they are perpendicular to each other.
Trigonometry Review
Opposite
Adjacent
Hypotenuse
Hypotenuse
Oppositesin
Hypotenuse
Adjacentcos
cos
sin
Adjacent
Oppositetan
Pythagorean Theorem:(Hypotenuse)2 = (Opposite)2 + (Adjacent)2
Adding Vectors by Components
If the components are perpendicular, they can be found using trigonometric functions.
sinVVy
cosVVx
V
Vy
V
Vx
cos
sin
Adj
Opptan
Hypotenuse
Oppositesin
Hypotenuse
Adjacentcos
Adding Vectors by Components
• The components are effectively one-dimensional, so they can be added arithmetically:
Adding Vectors by Components
Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use:
V
Vysin
Relative Velocity
•Relative velocity considers how observations made in different reference frames are related to each other.
Example: A person walks toward the front of a train at 5 km/h (VPT). The train is moving 80 km/h with respect to the ground (VTG). What is the person’s velocity with respect to the ground (VPG)?
TGPTPG VVV
hkmhkmhkmVPG /85/80/5
Relative Velocity
•Boat is aimed upstream so that it will move directly across.
•Boat is aimed directly across, so it will land at a point downstream.
•Can expect similar problems with airplanes.