1 / 27

Last Time : One-Dimensional Motion with Constant Acceleration Freely Falling Objects

Last Time : One-Dimensional Motion with Constant Acceleration Freely Falling Objects Wraps up our discussion of 1D motion … Today : Vector Techniques: Increasingly Important How to Manipulate Vectors Two-Dimensional Motion HW #2 due Thurs, Sept 9, 11:59 p.m.

daria-downs
Télécharger la présentation

Last Time : One-Dimensional Motion with Constant Acceleration Freely Falling Objects

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Last Time: One-Dimensional Motion with Constant Acceleration Freely Falling Objects Wraps up our discussion of 1D motion … • Today: Vector Techniques: Increasingly Important How to Manipulate Vectors Two-Dimensional Motion HW #2 due Thurs, Sept 9, 11:59 p.m. Recitation Quiz #2 tomorrow

  2. Vectors vs. Scalars • All of the physical quantities we will encounter in PHY 211 can be categorized as either a vector or a scalar. • Vectors have both a magnitude (size) and a direction. • Scalars have only a magnitude. Examples of Vectors Examples of Scalars Displacement Δx Speed Velocity Temperature (e.g., −40°F) Acceleration Mass Momentum Time Interval Angular Momentum Volumes, Areas “Vector Arithmetic” Ordinary Arithmetic

  3. Representing a Vector • We denote a vector mathematically in an equation as: a Example: scalar • We draw vectors as arrows • Points in the direction of the vector • Length indicates the magnitude. Magnitude is denoted with the variable name, without the arrow above it. B = magnitude of B A = magnitude of A

  4. Vector Equality • Two vectors A and B are equal if : • They have the same magnitude; and • They have the same direction. y These four vectors are all equal. A vector can be translated (or moved) parallel to itself without being affected. x

  5. Vector Addition: Geometric • Important: Make sure the vectors have the same units !! Geometrically: “Triangle Method of Addition” of A and B Draw A with its direction specified relative to some coordinate system. Draw B with the tail of B starting at the tip of A. The resultant vector R = A + B is the vector drawn from the tail of A to the tip of B. y x

  6. Commutative Law of Addition • Suppose have two vectors A and B. • Does A + B = B + A ? y y x x • Yes. Vector addition is “commutative”.

  7. Negative of a Vector • The negative of a vector A is defined to be the vector that when added to A, yields a resultant vector of 0. • Thus, A and –A have the same magnitude, but opposite directions. y A –A x

  8. Vector Subtraction • The operation A – B is defined to be y y x x

  9. Multiplying and Dividing by a Scalar • We can multiply or divide a vector by a scalar. (Recall, a scalar is just a number.) These processes yield a vector. c = scalar (number) c = scalar (number) Example: Example: A A Two times the magnitude, but in same direction !! Half of the magnitude, but in same direction !! B B

  10. Note • You cannot “multiply” or “divide” two vectors !

  11. Addition of > 2 Vectors • The same rules for vector addition apply in the addition of more than 2 vectors. Example: B B A A C C R

  12. Example • Vector A points 15 units in the +x direction. • Vector B points 15 units in the +y direction. • Find the magnitude and direction of: A – B

  13. Example • A car travels 50 km in the east direction, and then 100 km in the northeast direction. Using vectors, find the magnitude and direction of a single vector that gives the car’s displacement relative to its starting point. y, North 100 km 50 km 45° x, East

  14. Conceptual Question (p. 75) • If B is added to A, under what condition does the resultant vector have a magnitude equal to (A + B) ? i.e., the sum of the magnitudes of A and B

  15. Vector Components • The standard method of adding vectors makes use of the projections of the vectors along the axes of a coordinate system. These projections are called the components. • A vector can be completely specified by its components. “Components”: A =Ax+Ay Ax = A cosθ “component vectors” Ay = A sinθ Ay θ Ax • cosθ and sin θ determine the signs of Ax and Ay

  16. Example • A velocity vector has a magnitude of 10 m/s and a direction of 135° counter-clockwise from the +x-axis. Calculate the x- and y-components of this velocity vector.

  17. Example • A vector has an x-component of 5 units, and a y-component of –7 units. Find the magnitude and direction of the vector.

  18. Adding Vectors Algebraically • Suppose: R = A + B . The components of the resultant vector are given by: y y y + = B By + A Ay Ay By x x x Bx Ax + Bx Ax

  19. Adding Vectors Algebraically • Suppose: R = A + B . The components of the resultant vector are given by: y y y + = B By + A Ay Ay By x x x Bx Ax + Bx Ax

  20. Example: Problem 3.17 • The eye of a hurricane passes over an island in a direction of 60.0° north of west with a speed of 41.0 km/hr. • Three hours later, the hurricane suddenly shifts north, and its speed slows to 25 km/hr. • How far from the island is the hurricane 4.50 hours after it passes over the island? First 3 Hours: Travels 123 km @ 60° N of W 37.5 km Next 1.5 Hours: 123 km Travels 37.5 km @ N 60° Island

  21. Why Do We Need Vectors ? • Why did we spend all this time discussing vectors ? • In 1D-motion, vector quantities (velocity, acceleration) can be taken into account by specifying either a + or – sign. • In 2D- (or higher-dimensional) motion, this simple interpretation no longer works, and we must use vectors. 2D Example : 1D Example : If you make a series of steps along the x-axis, displacement from the origin is just the sum of the individual steps (taking account of signs). If you make a series of steps in the x-y plane, can’t just add up the magnitudes to find displacement from the origin! y x 0 x

  22. Displacement in 2D • In 2D, an object’s displacement is defined to be the change in its position vector : : position vector at time ti SI Unit: m : position vector at time tf y Object moving along this path in the x-y coordinate system Final position rf is just : initial displacement final x

  23. Velocity in 2D • In 2D, an object’s average velocity during a time interval Δt is its displacement divided by Δt: SI Unit: m/s This is a vector, just like the displacement ! • In 2D, an object’s instantaneous velocity is the limit of its average velocity as Δt becomes very small: SI Unit: m/s This is a vector, just like the displacement ! “value of the quantity as Δt becomes very small”

  24. Acceleration in 2D • In 2D, an object’s average acceleration during a time interval Δt is the change in its velocity divided by Δt: SI Unit: m/s2 This is a vector, just like the velocity ! • In 2D, an object’s instantaneous acceleration is the limit of its average acceleration as Δt becomes very small: SI Unit: m/s2 This is a vector, just like the displacement ! “value of the quantity as Δt becomes very small”

  25. Comment: Acceleration in 2D The acceleration is a vector, just like the velocity vector. If the magnitude of the velocity stays the same (speed), the velocity vector will still change if the direction changes ! In 2D, can have a non-zero acceleration if speed stays the same, but direction changes (example: driving in a circle at constant speed).

  26. Good News When dealing with vectors, we can usually break the vectors down into their x- and y-components. Motion in 2D can be then be thought of as two separate 1D problems, along the x- and y-axes.

  27. Reading Assignment • Next class: 3.4 – 3.5

More Related