Understanding Kinematics and Motion: Key Concepts and Applications

1. PHSX213 class • Questions from last time ? • Note: Labs today for some of you. • The web-site has a link to the eGradePlus resources. You should register and check this out. • Warm-ups may be posted there. • HW2 should appear soon. • Hand-outs: syllabus, course info., problem 3.24 solution, doing problems resources. • Describing Motion (Kinematics) • Chapter 2 leading into 4.

2. Basketball graph question.

3. Course information questions ?

4. Problem 3.24 solution

5. Problem solving framework

6. Kinematics: Special Case (1-D) Up-down Ball • Reference frame • Origin, (x, y, z)-axes • Position Vector, x = x i • Displacement, ≡ Dx = x2 – x1 • Instantaneous Velocity, v≡ d x /dt • Instantaneous Acceleration, a ≡ d v/dt ≡ d2 x /dt2 ^ → Draw graphs → → → → → → → →

7. What does all this mean ? • Position • Displacement (change in position over a time interval) • Velocity • Is synonymous (to me) with instantaneous velocity • average velocity, vave≡ Dr / Dt • Speed = magnitude of the velocity. • Acceleration • Instantaneous acceleration • average acceleration, aave≡ Dv / Dt • Can something traveling with constant speed be accelerating ?

8. Meter stick • Reaction time measurement

9. Draw graphs • 1) car (moving with constant speed) • 2) ball • 3) ball off ledge

10. Kinematic Equations • In general. • Will be useful for lab project, MX • Specific case of CONSTANT acceleration • v = v0 + a t • x – x0 = v0 t + ½ a t2 • v2 = v02 + 2 a (x – x0)

11. Free-Fall Acceleration • Falling Objects – Galileo • In the absence of air resistance, objects fall with the same acceleration, g, ≈ 9.8 m/s2 near the surface of the Earth. • We will return, to understand gravity in Ch. 13. • See Penny and Feather Video Experiment 2.2 • In vacuum, penny and feather hit ground at same time • Paper Demo • Glycerin Demo (will do this in next class) • Acceleration is not constant. Table 2.1 eqns don’t apply. • Use method of section 2.8 but with a(t).

12. Let’s say the dollar bill fell 16 cm before Jennifer tried to catch it. Calculate her reaction time. ( g = 9.8 m/s2 ) y – y0 = v0 t + ½ a t2 y0 = 0, v0 = 0, a = -g, y = -0.16 m So, t2 = - 2 y / g = (- 2 ) (-0.16) / (9.8) = 0.03265 s2 So, t = 0.1807 s = 0.18 s (only 2 sig. figs justified) Falling Dollar/Meter Stick/Reaction time Check dimensions Check if reasonable

13. Another constant acceleration example • Sample problem 2.7 is quite instructive. • A pitcher tosses a baseball up with an initial speed of 12 m/s • How long does the ball take to reach its maximum height ? • What is the ball’s maximum height ? • How long does it take to reach a height of 5.0 m ?

14. More about x, v, a graphs • From calculus, we know that • i) the slope of the graph is the first derivative. • So eg. x(t) vs t. The slope is dx/dt = v • v(t) vs t. The slope is dv/dt = a • ii) similarly, we know the area under the curve is the integral. • So eg. a(t) vs t. Area is ∫ a(t) dt = D v • v(t) vs t. Area is ∫ v(t) dt = D x

15. Which position-versus-time graph goes with this velocity-versus-time graph on the left? The particle’s position at ti = 0 s is xi = –10 m .

16. Which position-versus-time graph goes with this velocity-versus-time graph on the left? The particle’s position at ti = 0 s is xi = –10 m .

17. Position vector for a particle

18. Kinematics: General Case (3-D) • Reference frame • Origin, (x, y, z)-axes • Position Vector, r = x i + y j + z k • Displacement, ≡ Dr = r2 – r1 • Instantaneous Velocity, v≡ d r /dt • Instantaneous Acceleration, a ≡ d v/dt ≡ d2 r /dt2 ^ ^ ^ → → → → → → → → →

19. Next time • Projectiles • Uniform Circular Motion • Remember HW1 !