How do Sailboats Work? Vector Addition

Brandon Hoffman

General Physics

Houghton College

Outline

Background and Motivation

Theory

Experimental Procedure

Results

Conclusions

Outline

Why Sailboats?

Vector Addition

Sailcart Experiment

Outline

Why Sailboats?

Vector Addition

Sailcart Experiment

30°

AB

A + B

Outline

Why Sailboats?

30°

AB

A + B

Motivation

Practice using vectors as representations of forces.

Learn how vectors add together.

Improve trigonometry skills.

Vector Addition Applications

www.mit.eduwww. free-online-private-ground-school.com

www. Astronomyforbeginners.com

www. img.sparknotes.com

Theory

Ftotal = Ffan on cart + Ftrack on cart

= [(Ffan on cart + Ftrack on cart) cos()] i

+ [(Ffan on cart + Ftrack on cart)sin ()] j

= [(Ffan on cart + Ftrack on cart) cos()] i

= [(Ffan on cart) cos()] i

Forces on a Sailcart

30°

Fwind on cart

Ftrack on cart

Ftotal

Addition of Vectors

Fwind on cart Ftrack on cart

Ftotal

Ftotal = Fwind on cart + Ftrack on cart

= [Fwind on cart cos()] i

Procedure

• Angle between fan and sail kept constant at 30°

• Varied angle between sail and track.

• Measured acceleration of cart.

• Compared to theoretical prediction.

Apparatus

http://webapps.lsa.umich.edu

Apparatus

Apparatus

track

fan

sailcart

The Sailcart Experiment

30°

Angle between fan and sail held constant at 30

The Sailcart Experiment

30°

30°

30°30°

= 90°

30° = -30°

varied between -30 and 90

The Sailcart Experiment

30°

30°

30°30° 30

°

Ffan on cartF fan on cart

F fan

on ca

rt

30°

Ffan on cart

F fan

on c

art

Ffan on cart

Force of fan on cart always points forward.

Cart Mass (kg) Sail Angle (°) Cart Acceleration (cm/s2)

0.30 -30 34.6

0.30 0 40.0

0.30 30 34.6

0.30 60 20.0

0.30 75 10.4

0.30 90 0

0.60 -30 17.3

0.60 0 20.0

0.60 30 17.3

0.60 60 10.0

0.60 75 5.2

0.60 90 0

0.90 -30 11.5

0.90 0 13.3

0.90 30 11.5

0.90 60 6.7

0.90 75 3.5

0.90 90 0

Results

Conclusion

Cart Mass (kg) Sail Angle (°) Cart Acceleration (cm/s2)

0.30 -30 34.6

0.30 0 40.0

0.30 30 34.6

0.30 60 20.0

0.30 75 10.4

0.30 90 0

0.60 -30 17.3

0.60 0 20.0

0.60 30 17.3

0.60 60 10.0

0.60 75 5.2

0.60 90 0

0.90 -30 11.5

0.90 0 13.3

0.90 30 11.5

0.90 60 6.7

0.90 75 3.5

0.90 90 0

Results

Results

0

5

10

15

20

25

30

35

40

-30 -10 10 30 50 70 90

Results

Acceleration of Cart vs. Angle of Sail

0

5

10

15

20

25

30

35

40

-30 -10 10 30 50 70 90

Acceleration of Cart vs. Angle of Sail

0

5

10

15

20

25

30

35

40

-30 -10 10 30 50 70 90

Acceleration of Cart vs. Angle of Sail

0

5

10

15

20

25

30

35

40

-30 -10 10 30 50 70 90

Angle of Sail

Acc

eler

atio

n o

f C

art

Acceleration of Cart vs. Angle of Sail

0

5

10

15

20

25

30

35

40

-30 -10 10 30 50 70 90

Angle of Sail (degrees)

Acc

eler

atio

n o

f C

art

(cm

/s2)

0

5

10

15

20

25

30

35

40

-30 -10 10 30 50 70 90

Angle of Sail (degrees)

Acc

eler

atio

n o

f C

art

(cm

/s2)

Acceleration of Cart vs. Angle of Sail

Acceleration is directly proportional to cos().

Acceleration of Cart vs. Angle of Sail

Acceleration is directly proportional to cos ().

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

Cosine of Angle of Sail

Acc

eler

atio

n o

f C

art

(cm

/s2)

Conclusions

1. By adding the vectors representing the various forces together in head-to-tail fashion, we can calculate the total force on the sailcart. Because this force is proportional to the acceleration of the sailcart, the motion can be determined.

2. The acceleration of sailcart is directly proportional to the cosine of the angle between the sail and the direction of motion of the cart, assuming the angle between the fan and the sail is kept constant.

Conclusions

1. Vector addition can be used to predict sailboat motion.

2. Acceleration of sailcart has cosine dependence.

30°

Ffan on cart

Ftrack on cart

Ftotal

Acceleration of Cart vs. Angle of Sail

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

Cosine of Angle of Sail

Acc

eler

atio

n o

f C

art

(cm

/s2)

How do Sailboats Work? Vector Addition

Brandon Hoffman

General Physics

Houghton College

Outline

Why Sailboats?

30°

AB

A + B

Everybody Loves Sailing

Vector Addition Applications

www.mit.eduwww. free-online-private-ground-school.com

www. Astronomyforbeginners.com

www. img.sparknotes.com

Forces on a Sailcart

30°

Fwind on cart

Ftrack on cart

Ftotal

Addition of Vectors

Fwind on cart Ftrack on cart

Ftotal

Ftotal = Fwind on cart + Ftrack on cart

= [Fwind on cart cos()] i

Apparatus

sailcart

track

fan

The Sailcart Experiment

30°

The Sailcart Experiment

30°

30°

30°30°

= 90°

30° = -30°

The Sailcart Experiment

30°

30°

30°30° 30

°

Ffan on cartF fan on cart

F fan

on ca

rt

30°

Ffan on cart

F fan

on c

art

Ffan on cart

Force of fan on cart always points forward.

Acceleration of Cart vs. Angle of Sail

Acceleration is directly proportional to cos ().

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

Cosine of Angle of Sail

Acc

eler

atio

n o

f C

art

(cm

/s2)

Conclusions

1. Vector addition can be used to predict sailboat motion.

2. Acceleration of sailcart has cosine dependence.

30°

Ffan on cart

Ftrack on cart

Ftotal

Acceleration of Cart vs. Angle of Sail

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

Cosine of Angle of Sail

Acc

eler

atio

n o

f C

art

(cm

/s2)