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
 Kinematics is a part of dynamics
— In kinematics, you are interested in the description of motion
— Not concerned with the cause of the motion
Quantities in Motion
Any motion involves three concepts
— Displacement
— Velocity
— Acceleration
These concepts can be used to study objects in motion

Brief History of Motion

Sumaria and Egypt
— Mainly motion of heavenly bodies

Greeks
— Also to understand the motion of heavenly bodies
— Systematic and detailed studies
— Geocentric model
“Modern” Ideas of Motion
Copernicus
— Developed the heliocentric system

Galileo
— Made astronomical observations with a telescope
— Experimental evidence for description of motion
— Quantitative study of motion
Position
Defined in terms of a frame of reference
— A choice of coordinate axes
— Defines a starting point for measuring the motion
o Or any other quantity
— One dimensional, so generally the x- or y-axis
Displacement
Defined as the change in position
 o f stands for final and i stands for initial
— Units are meters (m) in SI
Displacement Examples
 From A to B

The displacement is positive, indicating the motion was in the positive x direction
 From C to F

The displacement is negative, indicating the motion was in the negative x direction

Displacement, Graphical
Vector and Scalar Quantities
Vector quantities need both magnitude (size) and direction to completely describe them
 Generally denoted by boldfaced type and an arrow over the letter
 + or — sign is sufficient for this chapter
Scalar quantities are completely described by magnitude only

Displacement Isn’t Distance

The displacement of an object is not the same as the distance it travels
Example: Throw a ball straight up and then catch it at the same point you released it
° The distance is twice the height
° The displacement is zero
Speed
The average speed of an object is defined as the total distance traveled divided by the total time
elapsed

Speed is a scalar quantity

Speed, cont
Average speed totally ignores any variations in the object’s actual motion during the trip
The path length and the total time are all that is important
 Both will be positive, so speed will be positive
SI units are m/s
Path Length vs. Distance
Distance depends only on the endpoints
— The distance does not depend on what happens between the endpoints
— Is the magnitude of the displacement
Path length will depend on the actual route taken

Velocity
It takes time for an object to undergo a displacement
The average velocity is rate at which the displacement occurs
Velocity can be positive or negative

— is always positive
Average speed is not the same as the average velocity

Velocity continued
Direction will be the same as the direction of the displacement, + or - is sufficient in one-
dimensional motion
Units of velocity are m/s (SI)
— Other units may be given in a problem, but generally will need to be converted to these
— In other systems:
 US Customary: ft/s
 cgs:cm/s
Speed vs. Velocity

Cars on both paths have the same average velocity since they had the same displacement in the
same time interval
The car on the blue path will have a greater average speed since the path length it traveled is
larger

Graphical Interpretation of Velocity

Velocity can be determined from a position time graph
Average velocity equals the slope of the line joining the initial and final points on the graph
An object moving with a constant velocity will have a graph that is a straight line

Average Velocity, Constant

The straight line indicates constant velocity

The slope of the line is the value of the average velocity

Notes on Slopes
The general equation for the slope of any line is

 The meaning of a specific slope will depend on the physical data being graphed
Slope carries units
Average Velocity, Non Constant
The motion is non-constant velocity
The average velocity is the slope of the straight line joining the initial and final points
Instantaneous Velocity
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 magnitude of the instantaneous velocity is what you read on a car’s speedometer

Instantaneous Velocity on a Graph

The slope of the line tangent to the position vs. time graph is defined to be the instantaneous
velocity at that time
 The instantaneous speed is defined as the magnitude of the instantaneous velocity

Graphical Instantaneous Velocity

Average velocities are the blue lines
The green line (tangent) is the instantaneous velocity
Acceleration
Changing velocity means an acceleration is present
Acceleration is the rate of change of the velocity

Units are m/s^2 (SI), cm/s^2 (cgs), and ft/s^2 (US Cust)

Average Acceleration
 Vector quantity
 When the object’s velocity and acceleration are in the same direction (either positive or
negative), then the speed of the object increases with time
 When the object’s velocity and acceleration are in the opposite directions, the speed of
the object decreases with time
Negative Acceleration
 A negative acceleration does not necessarily mean the object is slowing down
 If the acceleration and velocity are both negative, the object is speeding up
 ”Deceleration” means a decrease in speed, not a negative acceleration

Instantaneous and Uniform Acceleration

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
 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 vs. time graph
 Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time
graph

Average Acceleration — Graphical Example

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 (violet arrows maintain the same length)
 Velocity is increasing (red arrows are getting longer)
 Positive velocity and positive acceleration

Relationship Between Velocity and Acceleration

 Acceleration and velocity are in opposite directions

 Acceleration is uniform (violet arrows maintain the same length)
 Velocity is decreasing (red arrows are getting shorter)
 Velocity is positive and acceleration is negative

Motion Diagram Summary

Equations for Constant Acceleration

These equations are used in situations with uniform acceleration
Notes on the equations

 Gives displacement as a function of velocity and time

 Use when you don’t know and aren’t asked for the acceleration

 Shows velocity as a function of acceleration and time

 Use when you don’t know and aren’t asked to find the displacement

Graphical Interpretation of the Equation

Notes on the equations

  Gives displacement as a function of time, velocity and acceleration
 Use when you don’t know and aren’t asked to find the final velocity
 The area under the graph of v vs. t for any object is equal to the displacement of the
object

 Gives velocity as a function of acceleration and displacement

 Use when you don’t know and aren’t asked for the time
Problem-Solving Hints
 Read the problem
 Draw a diagram
o Choose a coordinate system
o Label initial and final points
o Indicate a positive direction for velocities and accelerations
 Label all quantities, be sure all the units are consistent
o Convert if necessary
 Choose the appropriate kinematic equation
 Solve for the unknowns
o You may have to solve two equations for two unknowns
 Check your results
o Estimate and compare
o Check units
Galileo Galilei
 1564 - 1642
 Galileo formulated the laws that govern the motion of objects in free fall
 Also looked at:
o Inclined planes
o Relative motion
o Thermometers
o Pendulum
Free Fall
 A freely falling object is any object moving freely under the influence of gravity alone
o Free fall does not depend on the object’s original motion
 All objects falling near the earth’s surface fall with a constant acceleration
 The acceleration is called the acceleration due to gravity, and indicated by g
Acceleration due to Gravity
 Symbolized by g
 g=9.80m/s^2
o When estimating, use g m 10 m/s^2
 g is always directed downward
o Toward the center of the earth
 Ignoring air resistance and assuming 9 doesn’t vary with altitude over short vertical
distances, free fall is constantly accelerated motion
Free Fall – an object dropped
 Initial velocity is zero
 Let up be positive
o Conventional
 Use the kinematic equations
o Generally use y instead of x since vertical
 Acceleration is g = -9.80 m/s^2

Free Fall – an object thrown downward

 a = g = -9.80 m/s^2
 Initial velocity
o With upward being positive, initial velocity will be negative
Free Fall – an object thrown upward
 Initial velocity is upward, so positive
 The instantaneous velocity at the maximum height is zero
 a = g = -9.80 m/s^2 everywhere in the motion
Thrown upward, cont.
The motion may be symmetrical
— Then t up = t down
— Then v = -v 0
The motion may not be symmetrical
— Break the motion into various parts
o Generally up and down
Non-symmetrical Free Fall Example
 Need to divide the motion into segments
 Possibilities include
o Upward and downward portions
o The symmetrical portion back to the release point and then the non-symmetrical
portion