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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
Resolving Vectors

  • Why?

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

  • How?

    • Use trig, of course!


Resolving vectors1

N

W

E

S

Resolving Vectors

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

25.0 m/s2

72º


Resolving vectors2

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 vectors3
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 vectors4
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
Adding Vectors

  • Two methods:

    • Graphically

      • Head-to-tail method

    • Mathematically


Head to tail method

N

W

E

R

S

Head-to-Tail Method

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

A

B


Head to tail method1

N

W

E

R

S

Head-to-Tail Method

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

B

A


Addition of vectors
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!


Addition of vectors1

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


Addition of vectors2

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


Addition of vectors3

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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