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

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt 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

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt 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…)

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt 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?

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt 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  ) Vectors become a lot easier to express and compute How are Cartesian and Polar coordinates related? O (0,0) (x 1,y 1 )=(r  ) r  y1y1 x1x1 +x +y

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt 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?

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt Difference between Speed and Velocity Let’s take a simple one dimensional translation that has many steps: Let’s call this line the X-axis Let’s have a couple of motions in a total time interval of 20 sec. +10m +15m -15m-5m-10m +5m Total Displacement: Total Distance Traveled: Average Velocity: Average Speed:

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt Example 2.1 Displacement: Average Velocity: Average Speed: 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 x 1 =50.0m to x 2 =30.5m, as shown in the figure. Find the displacement, distance, average velocity, and average speed. Distance: *Magnitudes of vectors are expressed in absolute values

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt 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. Here is where calculus comes in to help understanding the concept of “instantaneous quantities” Instantaneous speed is the size (magnitude) of the instantaneous velocity:

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt Position vs Time Plot time t1t1 t2t2 t3t3 t=0 Position x=0 x1x It is useful to understand motions to draw them on position vs time plots. Does this motion physically make sense? 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)

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt Example 2.3 (a) Determine the displacement of the engine during the interval from t 1 =3.00s to t 2 =5.00s. Displacement is, therefore: A jet engine moves along a track. Its position as a function of time is given by the equation x=At 2 +B where A=2.10m/s 2 and B=2.80m. (b) Determine the average velocity during this time interval.

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt Example 2.3 cont’d Calculus formula for derivative The derivative of the engine’s equation of motion is (c) Determine the instantaneous velocity at t=5.00s. and The instantaneous velocity at t=5.00s is

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt 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 Change of velocity in time (what kind of quantity is this?) Average acceleration: Instantaneous acceleration:

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt Meaning of Acceleration When an object is moving with a constant velocity ( v=v 0 ), 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? The answer is YES!! YES!

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt 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?

Monday January 26, PHYS , Spring 2004 Dr. Andrew Brandt Example 2.7 (a) Compute the average acceleration during the time interval from t 1 =3.00s to t 2 =5.00s. 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/s 2 ) t m. (b) Compute the particle’s instantaneous acceleration as a function of time. What does this mean? The acceleration of this particle is independent of time. This particle is moving under a constant acceleration.