Methods of Vector Addition Graphical & Mathematical Methods v1v1 v2v2 North East 2 km away ? ? ?

1 - Distinguish between vector and scalar quantities. 2 - Determine the sum of two vectors by a graphical method. 3 - Know how to represent vectors both graphically and mathematically. 4 - Be able to determine the magnitude and direction of a vector.

Scalars and Vectors Scalars are quantities which are fully described by a magnitude alone. Vectors are quantities which are fully described by both a magnitude and a direction Magnitude means how much or size or amount.

Scalar QuantitiesVector Quantities TimeDisplacement MassVelocity Distance (length)Acceleration SpeedForce EnergyMomentum TemperatureImpulse AreaTorque VolumeGravitational Field Density Angular Quantities of Velocity Acceleration momentum Pressure Work Power

Vectors must have a MAGNITUDE and DIRECTION –ex: displacement d = 10 mi North –ex: velocity v = 30 m/s at 30º –ex: accelerationa = - 10 m/s 2 Note that DIRECTION can be designated by a sign (+ or -), a cardinal direction (N, S, E, W), or an angle (0 o to 360 o ) Vectors

 Vectors are SCALED lines that represent physical phenomena that have both a MAGNITUDE (quantity) and a DIRECTION  Vectors may be added, subtracted, divided, or multiplied with each other.  Basic physics focuses on ADDING and SUBTRACTING vectors Vectors

 Vectors have a starting point (tail) and an ending point (head)  Vectors should be drawn to scale on graph paper, so that a vector with length of 4 is twice as long as a vector with a length of 2. tail head Vectors

The Length of a Vector is Proportional to It’s Magnitude 6 Centimeters 6 MPH A Vector Describing a Velocity of 6 MPH is Drawn With a Length of 6 Units 6 Inches Vectors

The magnitude of the speed is indicated by the measured length (scale) of the line. Vectors

Vectors Graphically Describe DIRECTION and MAGNITUDE One Foot/sec ++ = 3 Feet/sec And You can Add and subtract Them!

Direction is indicated by compass degrees, we use the bearing direction setup. 0o0o 90 o 180 o 270 o

 Vectors may be added in two basic ways ◦ Graphical Method ◦ Mathematical Method Vectors

The length of the line represents the quantity (distance or speed, etc.) and the arrow indicates the direction. Add vectors head to tail.

Point B Point C Point A Vectors Can Describe Displacement Total Displacement 12.2 miles North East of the Starting Position 10 miles East 7 miles North The Black Dashed Line is called the Resultant The magnitude of the Resultant Vector is found using the Pythagorean Theorem. B2B2 C2C2 A2A2 A 2 + B 2 = C 2

Ex: Find the net force of four people tugging on a rope. Sally pulls to the right with a force of 500 N, Billy to the right with 800 N, Keisha to the left with 750 N, and Jing to the left with 850 N. ∑ F = F Sally + F Billy + F Keisha + F Jing ∑ F = +500N + 800N - 750N - 850N ∑ F = N Vectors

Ex: Find the net force of four people tugging on a rope. Sally pulls to the right with a force of 500 N, Billy to the right with 800 N, Keisha to the left with 750 N, and Jing to the left with 850 N. Jing Keisha Sally Billy The length of the vector must be SCALED to represent the magnitude of the force Vectors

 Use a ruler to draw the vectors to the appropriate lengths and in the appropriate directions.  Vectors must be added head to tail JingKeisha SallyBilly Vectors

 A new vector is drawn between the tail of the first vector and the head of the last vector  This vector is called the RESULTANT and represents your answer.  Measure it with a ruler and note its direction. ∑F = N Vectors

 There are two different ways to add vectors graphically: ◦ Parallelogram method ◦ Head-to-tail method

Add 3 m/s East to 4m/s North  First, draw a dashed line parallel to the 3 E  Next, draw a dashed line parallel to the 4 N  The diagonal of the parallelogram is the RESULTANT East North v

 Add 3 m/s East to 4m/s North  First, draw a vector 3 units long going East  Next, add the tail of the second vector to the head of the first  Finally, connect head to tail with a new vector East North v

 Your new vector is called the RESULTANT.  Find the velocity (MAGNITUDE) of the resultant by measuring with a ruler and using your scale.  Find the angle (DIRECTION) of the resultant by using a protractor.

 A particle travels from A to B along the path shown by the dotted red line ◦ This is the distance traveled and is a scalar  The displacement is the solid line from A to B ◦ The displacement is independent of the path taken between the two points ◦ Displacement is a vector Vectors

 When adding vectors, their directions must be taken into account  Units must be the same  Graphical Methods ◦ Use scale drawings ◦ Accuracy difficult to control  Algebraic Methods ◦ Accuracy well defined Vectors

 Draw the first vector with the appropriate length and direction  Draw the next vector with the appropriate length and direction specified, whose origin is located at the end of vector  Continue drawing the vectors “tip-to-tail”  The resultant is drawn from the origin of to the end of the last vector  Measure the length and angle of Vectors

 When you have many vectors, just keep repeating the process until all are included  The resultant is still drawn from the origin of the first vector to the end of the last vector Vectors

 When two vectors are added, the sum is independent of the order of the addition. ◦ This is the commutative law of addition Vectors

Example: A car travels 3 km North, then 2 km Northeast, then 4 km West, and finally 3 km Southeast. What is the resultant displacement? A B C D R R is ~2.4 km, 13.5° W of N or °. D C A B

The absent minded professor leaving his physics lab for the day forgets where he parks his car. Walking across the campus he remembers he wrote down four displacement vectors: 2 blocks East, 3 blocks South, 4 blocks SouthEast and 2 blocks West. Where is his car?

Pack 14 of the Manassas Boy Scouts is hiking in the George Washington Forest. The head scout gives the pack some directions to where the camp is to be set. Their task is to find the camp based on the following: 50 km North, North of West, West of South and 20 km West. Find the Boy Scouts Camp with respect to where they Pack started.

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