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Vectors

Vectors. Learning objectives Scalars Versus Vectors The Components of a Vector Adding and Subtracting Vectors Unit Vectors Position, Displacement, Velocity, and Acceleration Vectors Relative Motion. This is one of the most important chapters in the course.

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Vectors

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  1. Vectors • Learning objectives • Scalars Versus Vectors • The Components of a Vector • Adding and Subtracting Vectors • Unit Vectors • Position, Displacement, Velocity, and Acceleration Vectors • Relative Motion This is one of the most important chapters in the course. PowerPoint presentations are compiled from Walker 3rd Edition Instructor CD-ROM and Dr. Daniel Bullock’s own resources

  2. Scalars and Vectors • Scalar  has only magnitude (size) • 5 seconds (time) • 10 kg (mass) • 13.6 eV (energy) • 10 m/s (speed) • Vector  has both magnitude (size) and direction • 10 m/s North (velocity) • - 5 m/s2 (acceleration) • 4.5 Newtons 370 North of West (force)

  3. Vectors • Vector has tail (beginning) and tip (end) • A position vector describes the location of a point • Can resolve vector into perpendicular components using a two-dimensional coordinate system: tip tail

  4. Vectors • Length, angle, and components can be calculated from each other using trigonometry: • Length (magnitude) of the vector A is given by the • Pythagorean theorem

  5. Vectors • Signs of vector components

  6. Adding and Subtracting Vectors • Adding vectors graphically: Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last. Adding Vectors Using Components: 1. Find the components of each vector to be added. 2. Add the x- and y-components separately. 3. Find the resultant vector.

  7. Adding and Subtracting Vectors • Subtracting Vectors: The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Here, D = A – B.

  8. Unit Vectors • Unit vectors are dimensionless vectors of unit length. • Multiplying unit vectors by scalars: the multiplier changes the length, and the sign indicates the direction.

  9. Vector Addition A motor boat is moving 15 km/hr relative to the water. The river current is 10 km/hr downstream. How fast does the boat go (relative to the shore) upstream and downstream? Boat Upstream Vector

  10. Vector Addition A motor boat is moving 15 km/hr relative to the water. The river current is 10 km/hr downstream. How fast does the boat go (relative to the shore) upstream and downstream? Boat Upstream Vector Boat Downstream Vector

  11. Vector Addition A motor boat is moving 15 km/hr relative to the water. The river current is 10 km/hr downstream. How fast does the boat go (relative to the shore) upstream and downstream? Boat Upstream Vector Boat Downstream Vector Current Vector = 10 km/hr downstream

  12. Boat Velocity Upstream Upstream: Place vectors head to tail, net result, 5 km/hr upstream

  13. Boat Velocity Upstream Upstream: Place vectors head to tail,

  14. Boat Velocity Upstream: Place vectors head to tail, net result, 5 km/hr upstream Start Finish Difference

  15. Boat Velocity Downstream: Place vectors head to tail,

  16. Boat Velocity Downstream: Place vectors head to tail, net result,

  17. Boat Velocity Downstream: Place vectors head to tail, net result, 25 km/hr downstream Commutative law

  18. Forces On An Airplane When will it fly? Gravity Propulsion Net Force?

  19. Forces On An Airplane When will it fly? Gravity Propulsion Net Force Plane Dives to the Ground

  20. Forces On An Airplane When will it fly? Gravity Propulsion Lift Net Force?

  21. Friction When will it fly? Gravity Propulsion Lift Net Force = 0 up or down Plane rolls along the runway like a car because of propulsion.

  22. Forces On An Airplane When will it fly? Gravity Propulsion Lift Net Force Plane Flies as long as Lift > Gravity

  23. Friction When will it fly? Gravity Propulsion Lift Air Resistance Net Force = 0 Equilibrium

  24. Flight When will it fly? Gravity Propulsion Lift Air Resistance Net Force Plane Flies as long as Lift > Gravity AND Propulsion > Air Resistance

  25. Adding (and subtracting) vectors by components Let’s say I have two vectors: I want to calculate the vector sum of these vectors: Let’s say the vectors have the following values:

  26. B A B Our result is consistent with the graphical method! A What’s the magnitude of our new vector?

  27. How would you find the angle, , the vector makes with the y-axis?  + B A opp = 2 adj = 12 

  28. Multiplying vectors by scalars: and it was multiplied by So if the vector A was: the scalar, a = 5 then the new vector: Scalar Product: (aka dot product): angle between the vectors mag. of a mag. of b

  29. Scalar Product: (aka dot product): vectors scalars • The dot product is the product • of two quantities: • mag. of one vector • scalar component of the • second vector along the • direction of the first

  30. Vector Product (aka cross product) The vector product produces a new vector who’s magnitude is given by: The direction of the new vector is given by the, “right hand rule” Mathematically, we can find the direction using matrix operations. The cross product is determined from three determinants

  31. 1st : Strike out the first column and first row! 2nd : Cross multiply the four components – and subtract: x - component The determinants are used to find the components of the vector 3rd : Strike out the 2nd column and first row 4th : Cross multiply the four components,subtract, and multiply by -1: y - component

  32. 5th: Cross out the last column and first row 6th : Cross multiply and subtract four elements z-component So then the new vector will be: We’ll look more at the scalar product when we talk about angular momentum. Example:

  33. Example: Notice the resultant vector is in the z – direction!

  34. Position, Displacement, Velocity, and Acceleration Vectors Average velocity vector: So vav is in the same direction as Δr.

  35. Position, Displacement, Velocity, and Acceleration Vector • Average acceleration vector is in the direction of the change in velocity:

  36. Position, Displacement, Velocity, and Acceleration Vector • Instantaneous velocity vector is tangent to the path:

  37. Position, Displacement, Velocity, and Acceleration Vector • Velocity vector is always in the direction of motion; acceleration vector can point anywhere:

  38. Relative Motion • The speed of the passenger with respect to the ground depends on the relative directions of the passenger’s and train’s speeds:

  39. Relative Motion This also works in two dimensions:

  40. Chapter 3 Summary • Scalar: number, with appropriate units • Vector: quantity with magnitude and direction • Vector components: Ax = A cos θ, By = B sin θ • Magnitude: A = (Ax2 + Ay2)1/2 • Direction: θ= tan-1 (Ay / Ax) • Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last

  41. Chapter 3 Summary • Component method: add components of individual vectors, then find magnitude and direction • Unit vectors are dimensionless and of unit length • Position vector points from origin to location • Displacement vector points from original position to final position • Velocity vector points in direction of motion • Acceleration vector points in direction of change of motion • Relative motion: v13 = v12 + v23

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