Chapter 2 Motion in One Dimension

Dynamics The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts Kinematics is a part of dynamics Kinematics is a part of dynamics In kinematics, you are interested in the description of motion In kinematics, you are interested in the description of motion Not concerned with the cause of the motion Not concerned with the cause of the motion

Brief History of Motion Sumaria and Egypt Sumaria and Egypt Mainly motion of heavenly bodies Mainly motion of heavenly bodies Greeks Greeks Also to understand the motion of heavenly bodies Also to understand the motion of heavenly bodies Systematic and detailed studies Systematic and detailed studies

“Modern” Ideas of Motion Galileo Galileo Made astronomical observations with a telescope Made astronomical observations with a telescope Experimental evidence for description of motion Experimental evidence for description of motion Quantitative study of motion Quantitative study of motion

Position Defined in terms of a frame of reference Defined in terms of a frame of reference One dimensional, so generally the x- or y- axis One dimensional, so generally the x- or y- axis

Vector Quantities Vector quantities need both magnitude (size) and direction to completely describe them Vector quantities need both magnitude (size) and direction to completely describe them Represented by an arrow, the length of the arrow is proportional to the magnitude of the vector Represented by an arrow, the length of the arrow is proportional to the magnitude of the vector Head of the arrow represents the direction Head of the arrow represents the direction Generally printed in bold face type Generally printed in bold face type

Scalar Quantities Scalar quantities are completely described by magnitude only Scalar quantities are completely described by magnitude only

Displacement Measures the change in position Measures the change in position Represented as  x (if horizontal) or  y (if vertical) Represented as  x (if horizontal) or  y (if vertical) Vector quantity Vector quantity + or - is generally sufficient to indicate direction for one-dimensional motion + or - is generally sufficient to indicate direction for one-dimensional motion Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in US Customary Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in US Customary

Displacements

Distance Distance may be, but is not necessarily, the magnitude of the displacement Blue line shows the distance Red line shows the displacement

Velocity It takes time for an object to undergo a displacement It takes time for an object to undergo a displacement The average velocity is rate at which the displacement occurs The average velocity is rate at which the displacement occurs generally use a time interval, so let t i = 0 generally use a time interval, so let t i = 0

Velocity continued Direction will be the same as the direction of the displacement (time interval is always positive) Direction will be the same as the direction of the displacement (time interval is always positive) + or - is sufficient + or - is sufficient Units of velocity are m/s (SI), cm/s (cgs) or ft/s (US Cust.) Units of velocity are m/s (SI), cm/s (cgs) or ft/s (US Cust.) Other units may be given in a problem, but generally will need to be converted to these Other units may be given in a problem, but generally will need to be converted to these

Speed Speed is a scalar quantity Speed is a scalar quantity same units as velocity same units as velocity total distance / total time total distance / total time May be, but is not necessarily, the magnitude of the velocity May be, but is not necessarily, the magnitude of the velocity

Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time The instantaneous velocity indicates what is happening at every point of time

Uniform Velocity Uniform velocity is constant velocity Uniform velocity is constant velocity The instantaneous velocities are always the same The instantaneous velocities are always the same All the instantaneous velocities will also equal the average velocity All the instantaneous velocities will also equal the average velocity

Graphical Interpretation of Velocity Velocity can be determined from a position- time graph Velocity can be determined from a position- time graph Average velocity equals the slope of the line joining the initial and final positions Average velocity equals the slope of the line joining the initial and final positions Instantaneous velocity is the slope of the tangent to the curve at the time of interest Instantaneous velocity is the slope of the tangent to the curve at the time of interest The instantaneous speed is the magnitude of the instantaneous velocity The instantaneous speed is the magnitude of the instantaneous velocity

Average Velocity

Instantaneous Velocity

Acceleration Changing velocity (non-uniform) means an acceleration is present Changing velocity (non-uniform) means an acceleration is present Acceleration is the rate of change of the velocity Acceleration is the rate of change of the velocity Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust) Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust)

Average Acceleration Vector quantity Vector quantity When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing

Instantaneous and Uniform Acceleration The limit of the average acceleration as the time interval goes to zero The limit of the average acceleration as the time interval goes to zero When the instantaneous accelerations are always the same, the acceleration will be uniform When the instantaneous accelerations are always the same, the acceleration will be uniform The instantaneous accelerations will all be equal to the average acceleration The instantaneous accelerations will all be equal to the average acceleration

Graphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph

Average Acceleration

Relationship Between Acceleration and Velocity Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero

Relationship Between Velocity and Acceleration Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer)

Relationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter)

Kinematic Equations Used in situations with uniform acceleration Used in situations with uniform acceleration

Notes on the equations Gives displacement as a function of velocity and time Gives displacement as a function of velocity and time

Notes on the equations Shows velocity as a function of acceleration and time Shows velocity as a function of acceleration and time

Graphical Interpretation of the Equation

Notes on the equations Gives displacement as a function of time, velocity and acceleration Gives displacement as a function of time, velocity and acceleration

Notes on the equations Gives velocity as a function of acceleration and displacement Gives velocity as a function of acceleration and displacement

Problem-Solving Hints Be sure all the units are consistent Be sure all the units are consistent Convert if necessary Convert if necessary Choose a coordinate system Choose a coordinate system Sketch the situation, labeling initial and final points, indicating a positive direction Sketch the situation, labeling initial and final points, indicating a positive direction Choose the appropriate kinematic equation Choose the appropriate kinematic equation Check your results Check your results

Free Fall All objects moving under the influence of only gravity are said to be in free fall All objects moving under the influence of only gravity are said to be in free fall All objects falling near the earth’s surface fall with a constant acceleration All objects falling near the earth’s surface fall with a constant acceleration Galileo originated our present ideas about free fall from his inclined planes Galileo originated our present ideas about free fall from his inclined planes The acceleration is called the acceleration due to gravity, and indicated by g The acceleration is called the acceleration due to gravity, and indicated by g

Acceleration due to Gravity Symbolized by g Symbolized by g g = 9.8 m/s² g = 9.8 m/s² g is always directed downward g is always directed downward toward the center of the earth toward the center of the earth

Free Fall -- an object dropped Initial velocity is zero Initial velocity is zero Let up be positive Let up be positive Use the kinematic equations Use the kinematic equations Generally use y instead of x since vertical Generally use y instead of x since vertical v o = 0 a = g

Free Fall -- an object thrown downward a = g a = g Initial velocity  0 Initial velocity  0 With upward being positive, initial velocity will be negative With upward being positive, initial velocity will be negative

Free Fall -- object thrown upward Initial velocity is upward, so positive Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero The instantaneous velocity at the maximum height is zero a = g everywhere in the motion a = g everywhere in the motion g is always downward, negative g is always downward, negative v = 0

Thrown upward, cont. The motion may be symmetrical The motion may be symmetrical then t up = t down then t up = t down then v f = -v o then v f = -v o The motion may not be symmetrical The motion may not be symmetrical Break the motion into various parts Break the motion into various parts generally up and down generally up and down

Non-symmetrical Free Fall Need to divide the motion into segments Need to divide the motion into segments Possibilities include Possibilities include Upward and downward portions Upward and downward portions The symmetrical portion back to the release point and then the non- symmetrical portion The symmetrical portion back to the release point and then the non- symmetrical portion

Combination Motions