Resolving vectors into components
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Resolving Vectors Into Components. And Vector Addition. Resolving Vectors. Why? Sometimes easier to analyze motion when we know its components. How? Use trig, of course!. N. W. E. S. Resolving Vectors. What are the North and East components of the vector shown at right?. 25.0 m/s 2.

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Resolving Vectors Into Components

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Resolving Vectors Into Components

And Vector Addition


Resolving Vectors

  • Why?

    • Sometimes easier to analyze motion when we know its components.

  • How?

    • Use trig, of course!


N

W

E

S

Resolving Vectors

  • 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

Resolving Vectors

  • 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


Resolving Vectors

  • 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


Resolving Vectors

  • 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


Adding Vectors

  • Two methods:

    • Graphically

      • Head-to-tail method

    • Mathematically


N

W

E

R

S

Head-to-Tail Method

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

A

B


N

W

E

R

S

Head-to-Tail Method

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

B

A


Addition of Vectors

  • 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

Addition of Vectors

  • 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

Addition of Vectors

  • 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

Addition of Vectors

  • 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


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