Chapter 2

Motion in One Dimension

Web Resources for Physics 1

• Physics Classroom

• http://www.khanacademy.org/science/physics

• http://ocw.mit.edu/courses/physics/

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

Aristotle (384 BC – 322 BC)

Aristotle

Added 5th element Ether

Aristotle added 5th Element - Ether

Aristotle on Motion

Aristotle’s classification of motion

• natural motion

– every object in the universe has a proper place determined by a combination of four elements: earth, water, air, and fire

– any object not in its proper place will strive to get there

examples:

• stones fall

• puffs of smoke rise

Aristotle on Motion

• natural motion (continued)

– straight up or straight down for all things on earth

– beyond Earth, motion is circular

example: Sun and Moon continually circle Earth

• violent motion

– produced by external pushes or pulls on objects

example: wind imposes motion on ships

Aristotle on Motion

Aristotle on Motion

Aristotle’s laws of motion

• The speed of falling is proportional to the weight of the object. Heavier objects fall faster.

• The speed by which an object falls depends inversely on the density of the medium it falls through (thus

there can be no void)

Galileo (1564– 1642)

Phases of the Venus

Galileo’s Concept of Inertia

Italian scientist Galileo demolished Aristotle’s assertions in early 1500s.

Galileo’s discovery • objects of different weight fall to the ground at the

same time in the absence of air resistance

• a moving object needs no force to keep it moving in the absence of friction

Position

• Defined in terms of a frame of reference

– One dimensional, so generally the x- or y-axis

– Defines a starting point for the motion

Displacement

• Defined as the change in position

–

• f stands for final and i stands for initial

– May be represented as y if vertical

– Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in US Customary

f ix x x

Displacements

Active Figure 2.2 (b), p.19

Table 2.1, p.21

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

• For example distance is a scalar

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

• Displacement has direction. Distance does not.

• Is displacement a vector or a scalar?

Figure 2.4, p.21

What is total displacement?

What is total distance?

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

total distanceAverage speed

total time

dv

t

Speed, cont

• Average speed totally ignores any variations in the object’s actual motion during the trip

• The total distance and the total time are all that is important

• SI units are m/s

Velocity

• It takes time for an object to undergo a displacement

• The average velocity is rate at which the displacement occurs

• generally use a time interval, so let ti = 0

• Is velocity a scalar or a vector?

f iaverage

f i

x xxv

t t t

Velocity continued

• Direction will be the same as the direction of the displacement (time interval is always positive) – + or - is sufficient

• 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

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 distance 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 positions

• 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

Average Velocity, Non Constant

• The motion is non-constant velocity

• The average velocity is the slope of the blue line joining two points

Table 2.1, p.21

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

lim

0t

xv

t

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

Uniform Velocity

• Uniform velocity is constant velocity

• The instantaneous velocities are always the same

– All the instantaneous velocities will also equal the average velocity

Acceleration

• Changing velocity (non-uniform) means an acceleration is present

• Acceleration is the rate of change of the velocity

• Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust)

• Is acceleration a scalar or a vector?

f i

f i

v vva

t t t

Average Acceleration

• 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 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

• 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-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)

• Positive velocity and positive acceleration

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)

• Velocity is positive and acceleration is negative

Kinematic Equations

• Used in situations with uniform acceleration

2

2 2

1

2

1

2

2

o

o

o

o

v v at

x vt v v t

x v t at

v v a x

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

t2

vvtvx fo

average

Notes on the equations

• Shows velocity as a function of acceleration and time

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

ov v at

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

2

o at2

1tvx

Notes on the equations

• Gives velocity as a function of acceleration and displacement

• Use when you don’t know and aren’t asked for the time

2 2 2ov v a x

Problem-Solving Hints

• Read the problem

• Draw a diagram – Choose a coordinate system, label initial and final points,

indicate a positive direction for velocities and accelerations

• Label all quantities, be sure all the units are consistent – Convert if necessary

• Choose the appropriate kinematic equation

Problem-Solving Hints, cont

• Solve for the unknowns

– You may have to solve two equations for two unknowns

• Check your results

– Estimate and compare

– Check units

Figure 2.14, p.30

Figure 2.15, p.31

Figure 2.16, p.32

Problem 4

• The Olympic record for the marathon is 2 h, 9 min, 21 s. The marathon distance is 26 mi, 385 yd. Determine the average speed (in miles per hour) of the record.

• Usain Bolt has the world record for the 100 m dash. 9.58 sec. What is this in miles per hour and how fast would he run a marathon at that rate

http://www.youtube.com/watch?v=3nbjhpcZ9_g

Fuel economy vs performance

Problem 18

• For velocity vs. time graph below:

(a) Find the average acceleration of the object during the time intervals 0 to 5.0 s, 5.0 s to 15 s, and 0 to 20 s. (b) Find the instantaneous acceleration at 2.0 s, 10 s, and 18 s.

Problem 27

• A drag racer starts her car from rest and accelerates at 10.0 m/s2 for a distance of 400 m (1/4 mile). (a) How long did it take the race car to travel this distance? (b) What is the speed of the race car at the end of the run?

Fastest production cars

Video

Problem 28

• A jet plane lands with a speed of 100 m/s and can accelerate at a maximum rate of –5.00 m/s2 as it comes to rest. (a) From the instant the plane touches the runway, what is the minimum time needed before it can come to rest? (b) Can this plane land on a small tropical island airport where the runway is 0.800 km long?

Galileo Galilei

• 1564 - 1642

• Galileo formulated the laws that govern the motion of objects in free fall

• Also looked at: – Inclined planes

– Relative motion

– Thermometers

– Pendulum

Free Fall

• All objects moving under the influence of gravity only are said to be in free fall

– 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 = 9.80 m/s² – When estimating, use g 10 m/s2

• g is always directed downward – toward the center of the earth

• Ignoring air resistance and assuming g 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

• Use the kinematic equations – Generally use y instead

of x since vertical

• Acceleration is g = -9.80 m/s2

vo= 0

a = g

Free Fall – an object thrown downward

• a = g = -9.80 m/s2

• Initial velocity 0

– With upward being positive, initial velocity will be negative

Free Fall -- object thrown upward

• Initial velocity is upward, so positive

• The instantaneous velocity at the maximum height is zero

• a = g = -9.80 m/s2 everywhere in the motion

v = 0

Thrown upward, cont.

• The motion may be symmetrical

– Then tup = tdown

– Then v = -vo

• The motion may not be symmetrical

– Break the motion into various parts

• Generally up and down

Non-symmetrical Free Fall

• Need to divide the motion into segments

• Possibilities include – Upward and downward

portions

– The symmetrical portion back to the release point and then the non-symmetrical portion

Combination Motions

Problem 34

• It is possible to shoot an arrow at a speed as high as 100 m/s. (a) If friction is neglected, how high would an arrow launched at this speed rise if shot straight up? (b) How long would the arrow be in the air?

Problem 52

• Another scheme to catch the roadrunner has failed! Now a safe falls from rest from the top of a 25.0-m-high cliff toward Wile E. Coyote, who is standing at the base. Wile first notices the safe after it has fallen 15.0 m. How long does he have to get out of the way?