Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda: Vectors Vectors in Mechanics Vector components

2. Reminder • MPHW1 (intro to Mastering Physics) due date is Friday 12pm

3. Motion in more than 1 dim • Have seen that 1d kinematics is written in terms of quantities with a magnitude and a sign • Examples of 1d vectors • To extend to d>1 need a more general definition of vector

4. Vectors • are used to denote quantities that have magnitude and direction • can be added and subtracted • can be multiplied or divided by a number • can be manipulated graphically (i.e., by drawing them out) or algebraically (by considering components)

5. Examples • Numbers: temperature, pressure, volume …. • Vectors: position, velocity, force • Vectors commonly denoted by boldface letters • Magnitude of A is written |A|

6. Adding vectors To add vector B to vector A: Draw vector A. Draw vector B with its tail starting from the tip of A. The sum vector A+B is the vector drawn from the tail of vector A to the tip of vector B.

7. Multiplying vector by number x 2.0 = x –1 = Note: A-B=A+(-1*B)

9. Find the vector “B – A” 1. 2. 3. 4.

10. Two unknown vectors A and B are added. The magnitude of the sum vector “A + B” (i.e., the quantity |A + B|) 1. is at least as great as |A| (i.e., the magnitude of A). 2. is at most as great as |A| + |B| (i.e., the magnitudes of A and B added). 3. must be equal to |A| + |B|. 4. can be greater than |A| + |B|.

11. 2D Motion Ds y s1 S – vector position Displacement is a vector! s2 O x

12. Velocity, Acceleration • If Ds is a vector then so is v=Ds/Dt • Similarly, so is the acceleration a=Dv/Dt

13. Ball rolling on incline velocity

14. Ball rolling on incline acceleration

15. Acceleration is not zero at top velocity time

16. Projection of a vector Component of a vector “How much a vector acts along some arbitrary direction” Projection onto one of the coordinate axes (x, y, z)

17. Components y A=Ax+Ay A A=axi+ayj Ay q i unit vector in x direction x Ax j unit vector in y direction ax,ay components of vector A Projection of A along coordinate axes

18. More components • Note (2D): ax=|A|cosq, ay=|A|sinq Or • Direction tanq=ay/ax • |A|2=ax2+ay2

19. Demo - projector and meterstick

20. Why components useful ? • Addition: just add components eg. if C=A+B cx=ax+bx; cy=ay+by • Subtraction similar • Multiply by number – just multiply components • Even more so in 3 (or higher ?!) dimensions

21. Quick Quiz Can any component of a vector ever be greater than the magnitude of the vector? 1. Yes 2. No 3. Not sure

22. Another Quick Quiz If one component of a vector is not zero, can the magnitude of the vector be zero? 1. Yes 2. No 3. Not sure

23. A bird is flying along a straight line in a direction somewhere East of North. After the bird has flown a distance of 2.5 miles, it is 2 miles North of where it started. How far to the East is it from its starting point? 1. 0 miles 2. 0.5 miles 3. 1.0 mile 4. 1.5 miles

24. 2D Motion in components Note: component of position vector along x direction is the x coordinate! y s – vector position s=xi+yj x

25. 2D Motion in components • x and y motions decouple • vx=Dx/Dt; vy=Dy/Dt • ax=Dvx/Dt; ay=Dvy/Dt • If acceleration is only non-zero in 1 direction – can choose coordinates so that 1 component of accel. is zero • Eg . motion under gravity

26. Simplest case • 2D motion with constant acceleration • Describes motion of ball under gravity (close to surface of Earth) • i.e x and y components of position vector satisfy const accel. equations …

27. Motion under gravity ax=0 vFx=vIx xF=xI+vIxt ay=-g vFy=vIy-gt yF=yI+vIyt-1/2gt2 y vIy=vsin(q) vIx=vcos(q) v q x

28. Projectile question • A ball is thrown at 450 to vertical with a speed of 7 m/s. Assuming g=10 m/s2 how far away does the ball land ?