Homework #1 due Thursday, Sept 2, 11:59 p.m. (via WebAssign) Last Time: Course Overview, Requirements Chapter 1: Units/Dimensions, Coordinate Systems, Trigonometry Reminder: Homework #1 due Thursday, Sept 2, 11:59 p.m. (via WebAssign) Today: Motion in One Dimension: Displacement, Speed, Velocity, Acceleration 11:00 p.m.  11:59 p.m. for all HWs (syllabus modified)

Recitation Quizzes Recitation Quiz #1 is tomorrow. Quiz will be given at the beginning of recitation. Be on time! Quizzes will consist of problem(s) that is(are) similar to: Concept(s) presented in the lectures; or Problem(s) on the current/recent HW assignment(s)

Kinematics vs. Dynamics The study of motion and of its causes (forces). Kinematics: The study of motion without regard to its causes. Today: Kinematics in one dimension (motion along straight line).

Displacement Suppose I walk in a straight line from one spot to another. What is needed to describe my motion quantitatively? First, need a coordinate system with a specified origin. x (m) 1 2 3 (origin) “Displacement” Δx defined to be the change in position : initial position : final position SI Unit : meter (m)

x (m) 1 2 3 (origin) If I walk from x = 1 m to x = 3 m, my displacement would be: x (m) 1 2 3 (origin) If I walk from x = 2.0 m to x = 0.5 m, my displacement would be: negative sign !!

Displacement Displacement Δx has both a magnitude (a size) and a direction (a positive or negative sign, telling you the direction). The displacement Δx is a vector quantity. Vectors have both a magnitude and a direction. A scalar quantity has only a magnitude (no direction). x (m) 1 2 3 (origin) Suppose turn around while walking : Vector, change in position Displacement is NOT a distance (scalar) ! Distance Traveled = 5 m

Speed vs. Velocity In everyday usage, you may use ‘speed’ and ‘velocity’ interchangeably. “The velocity of that Toyota Prius is 65 miles per hour.” In physics, this is technically not correct. Speed is a scalar quantity, having only a magnitude. Correct: “The speed of that Toyota Prius is 65 miles per hour.” Velocity is a vector quantity, having both a magnitude and a direction. Correct: “The velocity of that Toyota Prius is 65 miles per hour in the east direction.”

Average Speed The average speed of an object over a given time interval defined to be the total distance traveled divided by the total elapsed time [SI Unit: m/s] “is defined to be” With symbols, can be written as: In this context : “v” is average speed, not velocity d > 0 (always) and t > 0 (always)  v > 0 (always)

Example: Exercise 2.1 (p. 27) Calculate the average speed of the Apollo spacecraft in meters/second, given that the craft took 5 days to reach the Moon from Earth. (The Moon is 3.8  108 meters from Earth.) July 20, 1969

Average Velocity The average velocity v during a time interval Δt is the displacement Δx divided by Δt [SI Unit: m/s] xi is “initial position” at “initial time” ti xf is “final position” at “final time” tf Unlike average speed, average velocity is a vector quantity. Average velocity can be positive or negative (or even zero!) depending on the sign of Δx. (Δt is always positive) The sign of the average velocity does not necessarily have to to be the same as the average speed (which is always > 0).

Ex: Average Speed vs. Average Velocity Suppose at t = 0 s I start walking in a straight line, turn around, continue walking in a straight line, and then stop at t = 5 s. “initial” “final” x (m) 1 2 3 July 20, 1969 Calculate the average speed and the average velocity.

Graphical Interpretation of Velocity Suppose a bug flies to three different points along a straight line: A, B, and C Point t (s) x (m) A 2 B 3 4 C 5

Graphical Interpretation of Velocity Average Velocity During a Time Interval Δt = Slope of Straight Line Joining the Initial and Final Points During that Interval Δt x (m) Position vs. Time Graph B 4 In general, the average velocity between any distinct pair of points will be different. 3 2 A 1 C t (s) 1 2 3 4 5

Graphical Interpretation of Velocity Average Velocity During a Time Interval Δt = Slope of Straight Line Joining the Initial and Final Points During that Interval Δt x (m) Position vs. Time Graph B 4 In general, the average velocity between any distinct pair of points will be different. 3 2 A 1 C t (s) 1 2 3 4 5

Graphical Interpretation of Velocity Average Velocity During a Time Interval Δt = Slope of Straight Line Joining the Initial and Final Points During that Interval Δt x (m) Position vs. Time Graph B 4 In general, the average velocity between any distinct pair of points will be different. 3 2 A 1 C t (s) 1 2 3 4 5

Instantaneous Velocity One shortcoming of average velocity is that it cannot account for the details of what happens during an interval of time. If you are driving on I-75 between Lexington and Cincinnati, you would probably slow down, speed up, during the trip. At any time during the trip, your velocity at any particular time will not, in general, be equal to your average velocity over the entire trip. x In driving a car between two points: Average velocity must be computed over some period of time “Instantaneous velocity” can be determined at any instant of time from your speedometer (and knowing your direction of travel). t

Instantaneous Velocity The instantaneous velocity v is the limit of the average velocity as the time interval Δt becomes infinitesimally small: [SI Unit: m/s] x “value of (Δx/Δt) as Δt becomes very small” The instantaneous velocity is the slope of the line connecting the initial and final points for the short time interval Δt . Δx Δt t

Instantaneous Velocity For this example: x Δx2 Δt2 Δx1 Δt1 t

Instantaneous Velocity For this example: x Δx2 Green lines are the slopes of the lines “tangent” to the position vs. time curve at these particular times. Δt2 Δx1 Δt1 t

Instantaneous Velocity vs. Speed The instantaneous velocity is a vector. (Recall: it is nothing more than the average velocity, which is a vector, over a very short time interval Δt). The instantaneous speed is defined to be the magnitude of the instantaneous velocity. So it has no associated direction (i.e., it carries no algebraic sign, + or –). Example: Suppose Object 1 has an instantaneous velocity of +10 m/s, and Object 2 has an instantaneous velocity of –10 m/s. They both have the same instantaneous speed of 10 m/s.

Example: Problem 2.9(d) (d) Instantaneous velocity at t = 4.5 s ?

Acceleration Suppose a car is moving along a straight road: At time ti its velocity is vi At time tf its velocity is vf The average acceleration a during the time interval Δt = tf – ti is the change in velocity Δv divided by Δt SI Unit: m/s2 Example: ti = 0 s, vi = 0 m/s tf = 2 s, vf = 5 m/s

Acceleration Just like velocity, acceleration is a vector. It has both a magnitude, and a direction (for 1D motion, + or – sign). If the object’s velocity and acceleration are in the same direction, the object’s speed increases with time. If the object’s velocity and acceleration are in opposite directions, the object’s speed decreases with time. Example: Suppose vi = –5 m/s at ti = 0 s. If the average acceleration a = –10 m/s2 from ti = 0 s to tf = 5 s, what is vf at tf = 5 s ?

Instantaneous Acceleration Again, suppose you are driving from Lexington to Cincinnati. At certain times, your velocity is constant, and so your acceleration at that particular time is zero. At other times, when you pass a car, your velocity may increase, and so your acceleration is non-zero. It is useful to define an instantaneous acceleration, analogous to the instantaneous velocity. The instantaneous acceleration a is the limit of the average acceleration as the time interval Δt becomes infinitesimally small: SI Unit: m/s2

Instantaneous Acceleration Now, suppose we have a velocity-vs-time graph. Here, the velocity is increasing with time according to this red curve. Just like with the instantaneous velocity, the instantaneous acceleration is the slope of the line connecting the initial and final velocities for the time interval Δt. v Δv Δt t

Uniform Acceleration A special case of acceleration is that in which the velocity-vs-time graph forms a straight line. That is, the instantaneous acceleration is the same for all times. This is called uniform acceleration. This will be studied extensively in PHY 211. v Acceleration has a constant value. Slope is the same everywhere. t

Example 2.3 (p. 33) Instantaneous acceleration at A ?

Reading Assignment Next class: 2.5 – 2.6