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Resolving Vectors Into Components

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Resolving Vectors Into Components

And Vector Addition

- Why?
- Sometimes easier to analyze motion when we know its components.

- How?
- Use trig, of course!

N

W

E

S

- What are the North and East components of the vector shown at right?

25.0 m/s2

72º

N

E

25.0 m/s2

W

E

S

- East component:
- adjacent to angle

- cos72º =
- (25.0 m/s2)cos72º = E
- E = 7.73 m/s2 East

25.0 m/s2

72º

N

N

25.0 m/s2

W

E

S

Resolving Vectors

- North component:
- opposite from angle

- sin72º =
- (25.0 m/s2)sin72º = N
- N = 23.8 m/s2 North

25.0 m/s2

72º

- A cannon fires a shell with an initial velocity of 100. m/s at an angle of 25.0º above the horizontal. Find the vertical and horizontal components of the shell’s vo.

100. m/s

25.0º

- Horizontal velocity (h)
- h is adjacent to angle
- h = (100. m/s)(cos25.0º)
- h = 90.6 m/s

- Vertical velocity (v)
- v is opposite from angle
- v = (100. m/s)(sin25.0º)
- v = 42.3 m/s

- Two methods:
- Graphically
- Head-to-tail method

- Mathematically

- Graphically

N

W

E

R

S

- Sketch the resultant vector when vectors A and B are added.

A

B

N

W

E

R

S

- Sketch the resultant when vectors A and B are added.

B

A

- Two forces are acting on an object:
- 20.0 Newtons at an angle 20.0º N of E.
- 50.0 Newtons at an angle 50.0º N of E.

- Find the resultant force.
- Step 1: Resolve both vectors into their components.
- Step 2: Add the components.
- Step 3: Use trig and the Pyth.Theo. to find the resultant vector.
- Hint: Drawing a picture always helps!

N

B

A

E

- Resolve vector A:
- horizontal = 20.0 N(cos20º)
- horizontal = 18.8 N east
- vertical = 20.0 N(sin20º)
- vertical = 6.84 N north

- Resolve vector B:
- horizontal = 50.0 N(cos50º)
- horizontal = 32.1 N east
- vertical = 50.0 N(sin 50º)
- vertical = 38.3 N north

N

B

A

E

- Add horiz. and vert. components:
- Horizontal:
- 18.8 N east + 32.1 N east
- 50.9 N east

- Vertical:
- 6.84 N north + 38.3 N north
- 45.1 N north

N

B

E

A

R

- Magnitude of resultant:
- R = sqrt(N2 + E2)
- R = sqrt(45.12 + 50.92)
- R = sqrt(4620)
- R = 68.0 N

- Angle of resultant:
- tan = N/E
- tan = (45.1)/(50.9)
- tan = 0.886
- arctan0.886 = 41.6º N of E