AP Physics Mechanics for Physicists and Engineers Agenda

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AP Physics “Mechanics for Physicists and Engineers” Agenda for Today l l 1 -D

AP Physics “Mechanics for Physicists and Engineers” Agenda for Today l l 1 -D Kinematics (review). ç Average & instantaneous velocity and acceleration ç Motion with constant acceleration Introduction to calculus applications ç derivatives and slopes ç Integrals and area Physics II: Lecture 1, Pg 1

Kinematics Problems l l l 1 -D Kinematics ç Average & instantaneous velocity (Chapter

Kinematics Problems l l l 1 -D Kinematics ç Average & instantaneous velocity (Chapter 2 1, 4, 5, 11 -13, 15 -17) and acceleration (18, 21) Motion with constant acceleration(23, 24, 27, 31, 35, 37, 39, 401, 43) Free Fall (44, 47, 49, 51, 53, 56, 61, 63) Motion Graphs (66, 67, 69, 70) Review Phun!! Physics II: Lecture 1, Pg 2

Kinematics l l l Location and motion of objects is described using Kinematic Variables:

Kinematics l l l Location and motion of objects is described using Kinematic Variables: Some examples of kinematic variables. ç position r vector ç velocity v vector Kinematic Variables: ç Measured with respect to a reference frame. (x-y axis) ç Measured using coordinates (having units). ç Many kinematic variables are Vectors, Vectors which means they have a direction as well as a magnitude ç Vectors denoted by boldface V or arrow Physics II: Lecture 1, Pg 3

See text : 2 -1 Motion in 1 dimension l In general, position at

See text : 2 -1 Motion in 1 dimension l In general, position at time t 1 is usually denoted r(t 1). l In 1 -D, we usually write position as x(t 1 ). Since it’s in 1 -D, all we need to indicate direction is + or . l è Displacement in a time t = t 2 - t 1 is x = x(t 2 ) - x(t 1 ) = x 2 - x 1 x some particle’s trajectory in 1 -D x 2 x x 1 t 1 t t 2 t Physics II: Lecture 1, Pg 4

See text : 2 -1 1 -D kinematics l l Velocity v is the

See text : 2 -1 1 -D kinematics l l Velocity v is the “rate of change of position” Average velocity vav in the time t = t 2 - t 1 is: x x 2 x x 1 trajectory Vav = slope of line connecting x 1 and x 2. t 1 t t 2 t Physics II: Lecture 1, Pg 5

See text : 2 -2 1 -D kinematics. . . l Instantaneous velocity v

See text : 2 -2 1 -D kinematics. . . l Instantaneous velocity v is defined as: x so V(t 2 ) = slope of line tangent to path at t 2. x 2 x x 1 t 1 t t 2 t Physics II: Lecture 1, Pg 6

See text : 2 -3 1 -D kinematics. . . l Acceleration a is

See text : 2 -3 1 -D kinematics. . . l Acceleration a is the “rate of change of velocity” Average acceleration aav in the time t = t 2 - t 1 is: l And instantaneous acceleration a is defined as: l Physics II: Lecture 1, Pg 7

Recap l If the position x is known as a function of time, then

Recap l If the position x is known as a function of time, then we can find both velocity v and acceleration a as a function of time! x v a t t t Physics II: Lecture 1, Pg 8

More 1 -D kinematics l l l v 60 1 2 t We saw

More 1 -D kinematics l l l v 60 1 2 t We saw that v = x / t ç so therefore x = v t ( i. e. 60 mi/hr x 2 hr = 120 mi ) In “calculus” language we would write dx = v dt, which we can integrate to obtain: Graphically, this is adding up lots of small rectangles: v(t) + +. . . + = displacement t Physics II: Lecture 1, Pg 9

See text : 2 -4 1 -D Motion with constant acceleration l High-school calculus:

See text : 2 -4 1 -D Motion with constant acceleration l High-school calculus: l Also recall that l Since a is constant, we can integrate this using the above rule to find: l Similarly, since we can integrate again to get: Physics II: Lecture 1, Pg 10

See text : Table 2 -1 (p. 33) Recap l So for constant acceleration

See text : Table 2 -1 (p. 33) Recap l So for constant acceleration we find: x v l t From which we can derive: a t t Physics II: Lecture 1, Pg 11

Problem 1 l A car traveling with an initial velocity vo. At t =

Problem 1 l A car traveling with an initial velocity vo. At t = 0, the driver puts on the brakes, which slows the car at a rate of ab vo ab x = 0, t = 0 Physics II: Lecture 1, Pg 12

Problem 1. . . l A car traveling with an initial velocity vo. At

Problem 1. . . l A car traveling with an initial velocity vo. At t = 0, the driver puts on the brakes, which slows the car at a rate of ab. At what time tf does the car stop, and how much farther xf does it travel ? ? vo ab x = 0, t = 0 v=0 x = xf , t = tf Physics II: Lecture 1, Pg 13

Problem 1. . . l Above, we derived: (a) (b) l Realize that a

Problem 1. . . l Above, we derived: (a) (b) l Realize that a = -ab Using (b), realizing that v = 0 at t = tf : find 0 = v 0 - ab tf or tf = vo /af l Plugging this result into (a) we find the stopping distance: l Physics II: Lecture 1, Pg 14

Problem 1. . . l l l So we found that Suppose that vo

Problem 1. . . l l l So we found that Suppose that vo = 65 mi/hr x. 45 m/s / mi/hr = 29 m/s Suppose also that ab = |g| = 9. 8 m/s 2. ç Find that tf = 3 s and xf = 43 m Physics II: Lecture 1, Pg 15

Tips: l Read ! ç Before you start work on a problem, read the

Tips: l Read ! ç Before you start work on a problem, read the problem statement thoroughly. Make sure you understand what information in given, what is asked for, and the meaning of all the terms used in stating the problem. l Watch your units ! ç Always check the units of your answer, and carry the units along with your numbers during the calculation. l Understand the limits ! ç Many equations we use are special cases of more general laws. Understanding how they are derived will help you recognize their limitations (for example, constant acceleration). Physics II: Lecture 1, Pg 16

Recap of kinematics lectures l l 1 -D Kinematics ç Average & instantaneous velocity

Recap of kinematics lectures l l 1 -D Kinematics ç Average & instantaneous velocity (Chapter 3 - 1, 3, 7, 9, 11) and acceleration ç Motion Graphs (14, 15, 17, 19) Motion with constant acceleration(Ch 3 21, 23, 27, 29, 31, 35, 37 41) Free Fall (Ch 3 -41, 43, 47, 49, 51, 52 ) (Ch 3 Review Phun!! (67, 69, 70 ) ( Physics II: Lecture 1, Pg 17