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Lecture 4

Lecture 4. Today: Chapter 3 Introduce scalars and vectors Perform basic vector algebra (addition and subtraction) Interconvert between Cartesian & Polar coordinates. Finish. Start. q. q. 9.8 m. 9.8 m. An “interesting” 1D motion problem: A race.

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Lecture 4

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  1. Lecture 4 • Today: Chapter 3 • Introduce scalars and vectors • Perform basic vector algebra (addition and subtraction) • Interconvert between Cartesian & Polar coordinates

  2. Finish Start q q 9.8 m 9.8 m An “interesting” 1D motion problem: A race • Two cars start out at a speed of 9.8 m/s. One travels along the horizontal at constant velocity while the second follows a 9.8 m long incline angled q below the horizontal down and then an identical incline up to the finish point. The acceleration of the car on the incline is g sinq(because of gravity & little g) • At what angles q, if any, is the race a tie?

  3. A race • According to our dynamical equations • x(t) = xi + vi t + ½ a t2 • v(t) = vi + a t • A trivial solution is q = 0° and the race requires 2 seconds. • Horizontal travel • Constant velocity so t = d / v = 2 x 9.8 m (cosq) / 9.8 m/s • t1 = 2 cosq seconds • Incline travel • 9.8 m = 9.8 m/s ( t2 /2) + ½ g sinq (t2 /2)2 • 1 = ½ t2+ ½ sinq(½ t2)2 sinq (t2)2 + 4 t2- 8 =0

  4. Solving analytically is a challenge A sixth order polynominal, (if z=1, 1-1+4-8+4 =0)

  5. Solving graphically is easier Solve graphically q=0.63 rad q= 36° If q < 36°then the incline is faster If q < 36 °then the horizontal track is faster

  6. A science project • You drop a bus off the Willis Tower (442 m above the side walk). It so happens that Superman flies by at the same instant you release the bus. Superman is flying down at 35 m/s. • How fast is the bus going when it catches up to Superman?

  7. yi y 0 t A “science” project • You drop a bus off the Willis Tower (442 m above the side walk). It so happens that Superman flies by at the same instant you release the car. Superman is flying down at 35 m/s. • How fast is the bus going when it catches up to Superman? • Draw a picture

  8. yi y 0 t A “science” project • Draw a picture • Curves intersect at two points

  9. Coordinate Systems and Vectors • In 1 dimension, only 1 kind of system, • Linear Coordinates (x) +/- • In 2 dimensions there are two commonly used systems, • Cartesian Coordinates (x,y) • Circular Coordinates (r,q) • In 3 dimensions there are three commonly used systems, • Cartesian Coordinates (x,y,z) • Cylindrical Coordinates (r,q,z) • Spherical Coordinates (r,q,f)

  10. Scalars and Vectors • A scalar is an ordinary number. • Has magnitude ( + or - ), but no direction • May have units (e.g. kg) but can be just a number • Represented by an ordinary character Examples: mass (m, M) kilograms distance (d,s) meters spring constant (k) Newtons/meter

  11. A B C Vectors act like… • Vectors have both magnitude and a direction • Vectors: position, displacement, velocity, acceleration • Magnitude of a vector A≡ |A| • For vector addition or subtraction we can shift vector position at will (NO ROTATION) • Two vectors are equal if their directions, magnitudes & units match. A = C A ≠B, B ≠ C

  12. A A Vectors look like... • There are two common ways of indicating that something is a vector quantity: • Boldface notation: A • “Arrow” notation: Aor

  13. Scalars and Vectors • A scalar can’t be added to a vector, even if they have the same units. • The product of a vector and a scalar is another vector in the same “direction” but with modified magnitude B A = -0.75 B A

  14. While I conduct my daily run, several quantities describe my condition Which of the following is cannot be a vector ? Exercise Vectors and Scalars • my velocity (3 m/s) • my acceleration downhill (30 m/s2) • my destination (the lab - 100,000 m east) • my mass (150 kg)

  15. Vectors and 2D vector addition • The sum of two vectors is another vector. A = B + C B B A C C

  16. B - C -C B Different direction and magnitude ! B+C 2D Vector subtraction • Vector subtraction can be defined in terms of addition. = B + (-1)C B - C B -C

  17. U = |U| û û Unit Vectors • A Unit Vectorpoints : a length 1 and no units • Gives a direction. • Unit vector u points in the direction of U • Often denoted with a “hat”: u = û • Useful examples are the cartesian unit vectors [i, j, k] or • Point in the direction of the x, yand z axes. R = rx i + ry j + rz k or R = x i + y j + z k y j x i k z

  18. By C B Bx A Ay Ax Vector addition using components: • Consider, in 2D, C =A + B. (a) C = (Ax i + Ayj ) + (Bxi + Byj ) = (Ax + Bx )i + (Ay + By ) (b)C = (Cx i+ Cy j ) • Comparing components of (a) and (b): • Cx = Ax + Bx • Cy = Ay + By • |C| =[(Cx)2+ (Cy)2 ]1/2

  19. ExampleVector Addition • {3,-4,2} • {4,-2,5} • {5,-2,4} • None of the above • Vector A = {0,2,1} • Vector B = {3,0,2} • Vector C = {1,-4,2} What is the resultant vector, D, from adding A+B+C?

  20. tan-1 ( y / x ) Converting Coordinate Systems (Decomposing vectors) • In polar coordinates the vector R= (r,q) • In Cartesian the vectorR= (rx,ry) = (x,y) • We can convert between the two as follows: y (x,y) r ry  rx x • In 3D

  21. Motion in 2 or 3 dimensions • Position • Displacement • Velocity (avg.) • Acceleration (avg.)

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