PHYS 1443 – Section 501 Lecture #2

1. PHYS 1443 – Section 501Lecture #2 Monday January 26, 2004 Dr. Andrew Brandt • Chapter 2: One Dimensional Motion • Displacement • Velocity and Speed • Acceleration • Motion under constant acceleration PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

2. Announcements • Homework: 10 of you have signed up (out of ~40) • First homework assignment has been given on Ch. 2, due next Weds Feb. 4 • Remember! Homework counts 20% of the total • Two lowest HW’s dropped, but don’t make it the first two! • For lecture notes, viewing easiest with .pdf • To print could copy .ppt and print as hand out to 1x6 to save paper and avoid animation • Roll PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

3. Some Fundamentals • Kinematics: Description of Motion without understanding the cause of the motion • Dynamics: Description of motion accompanied by an understanding of the cause of the motion • Vector and Scalar quantities: • Scalar: Physical quantities that require magnitude but no direction • Speed, length, mass, height, volume, area, magnitude of a vector quantity, etc • Vector: Physical quantities that require both magnitude and direction • Velocity, Acceleration, Force, Momentum • It does not make sense to say “I ran with a velocity of 40 miles/hour.” • Objects can be treated as point-like if their sizes are smaller than the scale in the problem • Earth can be treated as a point like object (or a particle)in celestial problems • Simplification of the problem (The first step in setting up to solve a problem…) PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

4. Some More Fundamentals • Motion: Can be described as long as the position is known at any time (or position is expressed as a function of time) • Translation: Linear motion • Rotation: Circular or elliptical motion • Vibration: Oscillation • Dimensions: • 0 dimension: A point • 1 dimension: Linear drag of a point, resulting in a line  Motion in one-dimension is a motion in a line • 2 dimension: Linear drag of a line resulting in a surface • 3 dimension: Perpendicular Linear drag of a surface, resulting in a stereo object What about extra dimensions? PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

5. +y (x1,y1)=(r,q) y1 r q +x O (0,0) x1 Coordinate Systems • Makes it easy to express locations or positions • Two commonly used systems, depending on convenience • Cartesian (Rectangular) Coordinate System • Coordinates are expressed in (x,y) • Polar Coordinate System • Coordinates are expressed in (r,q) • Vectors become a lot easier to express and compute How are Cartesian and Polar coordinates related? PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

6. Displacement, Velocity and Speed One dimensional displacement is defined as: Displacement is the difference between initial and final positions of motion and is a vector quantity. How is this different from distance? Average velocity is defined as: Displacement per unit time through the total period of motion Average speed is defined as: What is the difference between speed and velocity? PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

7. Let’s call this line the X-axis +15m +5m +10m -5m -10m -15m Difference between Speed and Velocity • Let’s take a simple one dimensional translation that has many steps: Let’s have a couple of motions in a total time interval of 20 sec. Total Displacement: Average Velocity: Total Distance Traveled: Average Speed: PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

8. Example 2.1 The position of a runner as a function of time is plotted as moving along the x axis of a coordinate system. During a 3.00 s time interval, the runner’s position changes from x1=50.0m to x2=30.5m, as shown in the figure. Find the displacement, distance, average velocity, and average speed. • Displacement: • Distance: • Average Velocity: • Average Speed: *Magnitudes of vectors are expressed in absolute values PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

9. Here is where calculus comes in to help understanding the concept of “instantaneous quantities” Instantaneous Velocity and Speed Instantaneous velocity is defined as: • What does this mean? • Velocity in an infinitesimal time interval • Mathematically: Slope of the position variation as a function of time • An object undergoing a certain displacement might not move at the average velocity at all times. Instantaneous speed is the size (magnitude) of the instantaneous velocity: PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

10. Position x1 2 3 1 x=0 time Position vs Time Plot It is useful to understand motions to draw them on position vs time plots. t=0 t1 t2 t3 • 1. Running at a constant velocity (go from x=0 to x=x1 in t1, • displacement is + x1 in t1 time interval) x=ct -> dx/dt=v=c • 2. Velocity is 0 (go from x1 to x1 no matter how much time changes) • 3. Running at a constant velocity but in the reverse direction as 1. • (go from x1 to x=0 in t3-t2 time interval, displacement is - x1 in • t3-t2 time interval) Does this motion physically make sense? PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

11. A jet engine moves along a track. Its position as a function of time is given by the equation x=At2+B where A=2.10m/s2 and B=2.80m. Example 2.3 (a) Determine the displacement of the engine during the interval from t1=3.00s to t2=5.00s. Displacement is, therefore: (b) Determine the average velocity during this time interval. PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

12. Example 2.3 cont’d (c) Determine the instantaneous velocity at t=5.00s. Calculus formula for derivative and The derivative of the engine’s equation of motion is The instantaneous velocity at t=5.00s is PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

13. Acceleration Change of velocity in time (what kind of quantity is this?) • Average acceleration: • In calculus terms: A slope (derivative) of velocity with respect to time or change of slopes of position as a function of time analogous to • Instantaneous acceleration: analogous to PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

14. Meaning of Acceleration • When an object is moving with a constant velocity (v=v0), there is no acceleration (a=0) • Could there be acceleration when an object is not moving? • When an object is moving faster as time goes on, (v=v(t) ), acceleration is positive (a>0) • When an object is moving slower as time goes on, (v=v(t) ), acceleration is negative (a<0) • Is there acceleration if an object moves in a constant speed but changes direction? YES! The answer is YES!! PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

15. Example 2.4 A car accelerates along a straight road from rest to 75km/h in 5.0s. What is the magnitude of its average acceleration? PHYS 1443-501, Spring 2004 Dr. Andrew Brandt

16. Example 2.7 A particle is moving in a straight line so that its position as a function of time is given by the equation x=(2.10m/s2)t2+2.8m. (a) Compute the average acceleration during the time interval from t1=3.00s to t2=5.00s. (b) Compute the particle’s instantaneous acceleration as a function of time. The acceleration of this particle is independent of time. What does this mean? This particle is moving under a constant acceleration. PHYS 1443-501, Spring 2004 Dr. Andrew Brandt