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Physics 218

Lecture 3: Kinematics. Physics 218. Alexei Safonov. Today. Finish material from last time Start Kinematics. z. ^. k. y. x. ^. ^. j. i. First Question:. A . B = |A||B|Cos Q. Harder Example. Vector Cross Product. This is the last way of multiplying vectors we will see

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Physics 218

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  1. Lecture 3: Kinematics Physics 218 Alexei Safonov

  2. Today • Finish material from last time • Start Kinematics

  3. z ^ k y x ^ ^ j i First Question: A.B = |A||B|CosQ

  4. Harder Example

  5. Vector Cross Product • This is the last way of multiplying vectors we will see • Direction from the “right-hand rule” • Swing from A into B!

  6. z ^ k y x ^ ^ j i _ + Vector Cross Product Cont… • Multiply out, but use the Sinq to give the magnitude, and RHR to give the direction

  7. Cross Product Example

  8. Checklist for Today • Things that were due at various points in time before you came here today: • Mastering physics: homework for Ch.1 • Pre-lecture (SmartPhysics) for Ch. 2 and the checkpoints • Reading for Chapter 2 • Should have started homework for Chapter 2 • For this week: • Be ready for the lab (WebAssign) including pre-labs and all • Quiz at the recitation on Chapter 1 • No pre-lecture for the Wednesday lecture (this week only, this is not typical) • Homework for Chapter 2 is due Sunday on MP • Pre-lectures for next week due on Sunday

  9. Pre-Lecture Participation • Total participation: 134/149 • 15 people have already lost potential points and risk falling behind • If you are in the first 2-3 bins (except a few with strong past background for whom this material was very easy), you are wasting your time and money in this class

  10. Kinematics: Describing Motion Interested in two key ideas: • How objects move as a function of time • Kinematics • Chapters 2 and 3 • Why objects move the way they do • Dynamics • Do this in Chapter 4 and later

  11. Chapter 2: Motion in 1-Dimension • Velocity & Acceleration • Equations of Motion • Definitions • Some calculus (derivatives) • Wednesday: • More calculus (integrals) • Problems

  12. Notes before we begin • This chapter is a good example of a set of material that is best learned by doing examples • We’ll do some examples today • Lots more next time…

  13. Equations of Motion We want Equations that describe: • Where am I as a function of time? • How fast am I moving as a function of time? • What direction am I moving as a function of time? • Is my velocity changing? Etc.

  14. Motion in One Dimension • Where is the car? • X=0feet at t0=0 sec • X=22feet at t1=1sec • X=44feet at t2=2sec • We say this car has “velocity” or “Speed” • Plot position vs. time. How do we get the velocity from the graph?

  15. Motion in One Dimension Cont… • Velocity:“Change in position during a certain amount of time” • Calculate from the Slope: The “Change in position as a function of time” • Change in Vertical • Change in Horizontal • Change: D • Velocity DX/Dt

  16. Constant Velocity • Equation of Motion for this example: • X = bt • Slope is constant • Velocity is constant • Easy to calculate • Same everywhere

  17. Moving Car A harder example: X = ct2 What’s the velocity at t=1 sec? Want to calculate the “Slope” here. Instantaneous Velocity What would my speedometer read?

  18. Displacement and Velocity in One Dimension Displacement Time taken

  19. Displacement and Velocity in One Dimension Thev(t)vs.tplot is just theslope of thex(t)vs.tplot Definition: Speed = |v(t)|

  20. Math: Derivatives • To find the slope at time t, just take the “derivative” • For X=ct2, Slope = V =dx/dt =2ct • Derivative for most common case: • If x= atn V=dx/dt=natn-1 • “Derivative of X with respect to t” • More examples • X= qt2 V=dx/dt=2qt • X= ht3 V=dx/dt=3ht2

  21. Derivatives Math • Basic rules: • Sum: x(t)= f(t) + g(t) => dx/dt = df/dt+dg/dt • x(t)= 12+6t+3t2+4t3 => dx/dt =0+6+6t+12t2 • Constant: x(t)= C => dx/dt = 0 • Multiplication: x(t)=f(t)*g(t) => dx/dt = df/dt * g + f * dg/dt • x(t)= (2t)*(t2)=> dx/dt =2*(t2)+(2t)*(2t)=6t2 • Compare to x(t)= (2t)*(t2)=2t3 => dx/dt=2*3*t2=6t2

  22. Common Mistakes • The trick is to remember what you are taking the derivative “with respect to” • More Examples (with a=constant): • What if X= 2a3tn? • Why not dx/dt = 3(2a2tn)? • Why not dx/dt = 3n(2a2tn-1)? • What if X= 2a3? • What is dx/dt? • There are no t’s!!! dx/dt = 0!!! • If X=22 feet, what is the velocity? =0!!!

  23. Check: Constant Position • X = C = 22 feet • V = slope = dx/dt = 0

  24. Check: Constant Velocity • Car is moving • X=0 feet at t0=0 sec • X=22feet at t1=1 sec • X=44feet at t2=2 sec • What is the equation of motion? • X = bt with b=22ft/sec • V = dX/dt V= b = 22ft/sec

  25. Check: Non-Constant Velocity • X = ct2 with c=11ft/sec2 • V = dX/dt = 2ct • The velocity is: • “non-Constant” • a “function of time” • “Changes with time” • V=0ft/s at t0=0 sec • V=22ft/s at t1=1 sec • V=44 ft/s at t2=2sec

  26. Acceleration • If the velocity is changing, we are “accelerating” • Acceleration is the “Rate of change of velocity” • Just like velocity is the rate of change in distance (or position) • You hit the accelerator in your car to speed up • (It’s true you also hit it to stay at constant velocity, but that’s because friction is slowing you down…we’ll get to that later…)

  27. Acceleration • Acceleration is the “Rate of change of velocity” • Said differently: “How fast is the Velocity changing?” “What is the change in velocity as a function of time?”

  28. Acceleration

  29. Checkpoint 1 • For the Displacement and Velocity curves shown on the left, which is the correct plot of acceleration vs. time? A B

  30. Checklist for Today • Things that were due at various points in time before you came here today: • Mastering physics: homework for Ch.1 • Pre-lecture (SmartPhysics) for Ch. 2 and the checkpoints • Reading for Chapter 2 • Should have started homework for Chapter 2 • For this week: • Be ready for the lab (WebAssign) including pre-labs and all • Quiz at the recitation on Chapter 1 • No pre-lecture for the Wednesday lecture (this week only, this is not typical) • Homework for Chapter 2 is due Sunday on MP • Pre-lectures for next week due on Sunday

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