Descriptions of motion involve distance and speed simple numbers w units called scalars Reference Frames Speed Often represent reference frames as Cartesian or rectangular drawings Can combine distance change in position and duration change in time to create a quantity called speed In m s Distance and duration are scalars as well as speed Speed is 0 if an object remains at rest Average speed v v distance covered duration of trip d t Instantaneous speed the value of speed at any point instant in time Can change and may not affect the average speed calculation Not the same as v as duration becomes small the two become the same number Vectors Vectors carry 2 pieces of into they contain magnitudes scalars and directions Displacement combines the scalar distance with direction Velocity combines the scalar speed with direction We ll deal with negative vectors This doesn t mean that the magnitude of the vector is negative it refers to the direction The vector a arrow over pointing to right has the same magnitude as the vector a arrow over pointing to right Tail head Head tail Vectors are added head to tail a a 0 The result vector goes from the tail of the first to the head of the second Displacement a a Distance covered a distance covered a Total distance covered 2a Total displacement is 0 Velocity Velocity vector points in the direction of the displacement vector Need to take direction into account when calculating average velocity During round trip travel your distance is not 0 but your displacement is this makes the average speed and velocity differ Reading Notes Acceleration Acceleration is the rate of change of velocity with time An object accelerates whenever its velocity changes no matter what the change Average acceleration is the change in velocity divided by the change in time increase or decrease Aav v t v f vi tf ti SI unit meter per second per second m s2 Instantaneous Acceleration Vector When acceleration is constant the instantaneous and average accelerations are the same Relating the Signs of Velocity and Acceleration to the Change in Speed In one dimension nonzero velocities and accelerations are either or depending on whether they point in the positive or negative direction of the coordinate system chosen Velocity and acceleration of an object may have the same or opposite signs Two possibilities in one dimension When the velocity and acceleration of an object have the same sign the speed of the object increases When the velocity and acceleration of an object have opposite signs the speed of an object decreases Lecture Kinematics Graphical Analysis v final pos initial pos final time initial time This expression is just that for the slope of a straight line slope rise run Average velocity is a slope or rate of change Kinematics Points to Take Away Acceleration is a vector quantity that tells you how velocity changes with time Just as position vs time plots Acceleration A change in velocity with time Not caused by velocity and velocity is not caused by acceleration Average acceleration is the slope of a velocity vs time curve What does it mean to have a negative acceleration The instantaneous acceleration Since tfinal is always larger than tinitial it means that vfinal must be more negative than vinitial 2 5 2 7 Notes Motion with Constant Acceleration Aav vf vi tf ti a v v0 a t 0 at Constant Acceleration Equation of Motion Velocity as a Function of Time v v0 at Describes a straight line on a v versus t plot Crosses the velocity axis at the value v0 and has a slope a in agreement with the graphical interpretations discussed in section 2 4 Position as a Function of Time and Velocity x x0 vavt Constant Acceleration Equation of Motion Average Velocity v av v0 v Acceleration isn t constant Constant Acceleration Equation of Motion Position as a Function of Time x x0 v0 v t Freely Falling Objects Motion with constant acceleration could be free fall Objects of different weight fall with the same constant acceleration Provided air resistance is small enough to be ignored Characteristics of Free Fall If in free fall it is assumed that an object s motion is not influenced by any form of friction or air resistance Any motion under the influence of gravity alone An object is in free fall as soon as it is released whether it is dropped from rest thrown downward or thrown upward Acceleration produced by gravity on the Earth s surface is denoted with the symbol g g the acceleration due to gravity g 9 81 m s 2 Lecture Notes Linear Motion 3 1 3 4 Scalars Versus Vectors Lecture Points to Take Away 1 D linear motion dealing w vectors is easy there is only one direction and motion is either in the or direction A scalar is a number with units It can be or 0 Sometimes a direction is also needed though A vector is a mathematical quantity with both a magnitude and a direction TO INDICATE A VECTOR WITH A WRITTEN SYMBOL WE USE BOLDFACE FOR THE VECTOR ITSELF WITH A SMALL ARROW ABOVE IT TO REMIND US OF ITS VECTOR NATURE AND WE USE ITALICS TO INDICATE ITS MAGNITUDE Motion in two dimensions can be broken down into two independent directions Vectors have magnitudes and directions 2 D Reference Frames In two dimensions I need two numbers to specify the location of an object Latitude and longitude X and y Vectors in 2 D All the vectors that have been discussed behave the same way displacement velocity and acceleration Any 2D vector can be broken down into an x component and a y component How would you handle a 3D vector Vector Arithmetic Vectors can be added and subtracted 3 5 3 6 Position vector Relative Velocity We often define our reference frame to be the surface of the Earth We can always decide that we are the one that is not moving and consider everything This means that we consider the surface of the EArth to be unmoving else including the surface of the earth to be moving Our choice of reference frame cannot change reality We could choose to follow the motion of a ball in free fall in a reference frame attached to the ball but it wouldn t change the fact that the ball encounters the surface of the Earth 1D with one Reference Frame Velocities first have to be defined in some initial reference frame with respect to the ground Same information present but it s appearance has changed The relative velocity of one object with respect to another object is the difference between their velocities as measured in the reference frame Descriptions of the vectors become more complicated because we have to