As a first step in studying classical mechanics, we describe motion in terms of space and time while ignoring the agents that caused that motion. This portion of classical mechanics is called kinematics.

CHAPTER 2:

MOTION IN ONE DIMENSION

*In this chapter, we consider only motion in one dimension, that is, motion along a straight line.

*In that translational motion, we describe the moving object as a particle regardless of its size.

Chapter outline

2.1 Position, Velocity, and Speed2.2 Instantaneous Velocity and Speed2.3 Acceleration2.4 Motion Diagrams2.5 One dimensional Motion with Constant Acceleration2.6 Freely Falling Objects2.7 Kinematic Equations Derived from CalculusSummary

2.1 Position, Velocity, and Speed:

a. Position:

-A particle’s position is the location of the particle with respect to a chosen reference point that we can consider to be the origin of a coordinate system.

-The motion of a particle is completely known if the particle’ s position in space is known at all times.

Figure 2.1 (a): 1 chiếc xe - mà ta xem như một chất điểm - chuyển động tới lui dọc theo đường thẳng trục Ox.

Figure 2.1 (b): Position – time graph for the motion of the particle.

b. Velocity:

-Displacement of a particle: is it’s change in position in some time interval.

When it moves from an initial position to a final position , the displacement of a particle is given by:

2.1f ix x x

ix

fx

0 :

.

0 :

.

f i

f i

x x x this case corresponds to

a particle moving in the positive x direction

x x x this case corresponds to

a particle moving in the negative x direction

Note that: it is very important to recognize the difference between displacement and distance traveled:

+ Displacement of a particle: is it’s change in position in some time interval. Displacement is a vector quantity.

+ Distance is the length of a path followed by a particle. Distance is a scalar quantity.

- Average velocity of a particle: is defined as the particle’s displacement divided by the time interval during which that displacement occurs:

x t

2.2x

xv

t

The average velocity of a particle moving in one dimension can be positive or negative, depending on the sign of the displacement.

, 0 0

, 0 0

f i x

f i x

xif x x then x and v

tx

if x x then x and vt

*Interpreting average velocity geometrically: We draw a straight line between any two points on the position – time graph in figure 2.1b. This line forms the hypotenuse of a right triangle of height and base . The slope of this line is the ratio which is what we have defined as average velocity .

x t/x t

For example: the line between positions A and B in figure 2.1b has slop equal to the average velocity of the car between those two times:

(52 - 30)/(10 - 0) = 2.2 m/s

c.The average speed:

-The average speed of a particle, a scalar quantity: is defined as the total distance traveled divided by the total time interval required to travel that distance:

tan2.3

total dis ceAverage speed

total time

*Notice: the distinction between average velocity and average speed. + average velocity is the displacement divided by the time interval. + average speed is the distance divided by the time interval.

2.2 Instantaneous Velocity and Speed:

Figure 2.3 (b): An enlargement of the upper – left – hand corner of the graph shows how the blue line between posision A and B approaches the green tangent line as point B is moved closer to point A.

2.2 Instantaneous Velocity and Speed:

a. The Instantaneous Velocity :

- The Instantaneous Velocity equals the limiting value of the ratio as approaches zero:

xv

/x t t

0

lim 2.4x t

xv

t

- In calculus notation, this limit is called the derivative of x with respect to t, written dx/dt:

0

lim 2.5xt

x dxv

t dt

*Notice:

+ The instantaneous velocity can be positive, negative, or zero.

+ From here on, we use the word velocity to designate instantaneous velocity. When it is average velocity we are interested in, we shall always use the adjective average.

b. Instantaneous speed:

-The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity.-Instantaneous speed has no direction associated with it and hence carries no algebraic sign.

Example: if one particle has an instantaneous velocity of + 25m/s along a given line and another particle has an instantaneous velocity of – 25m/s along the same line, both have a speed of 25m/s.

*Notice:

+ The magnitude of the average velocity is not the average speed.

+ The difference when discussing instantaneous values: that is, the magnitude of the instantaneous velocity is the instantaneous speed.

+ In an infinitesimal time interval, the magnitude of the displacement is equal to the distance traveled by the particle.

2.3 Acceleration:

a. Average acceleration:

-The average acceleration of the particle is defined as the change in velocity divided by the time interval during which that change occurs:

xa

xvt

2.6xf xixx

f i

v vva

t t t

b. Instantaneous acceleration:

-The instantaneous acceleration is defined as the limit of the average acceleration as approaches zero. t

0

lim 2.6xx t

va

t

- In calculus notation, the instantaneous acceleration equals the derivative of the velocity with respect to time.

2

20lim 2.7x x

xt

v dv d xa

t dt dt

*Interpreting accleration geometrically: If the velocity of a moving particle is the slope of the particle’ s position – time graph, the acceleration of a particle is the slope of the particle’ s velocityv- time graph.

Figure 2.5: (b) Velocity – time graph for the car (particle) moving in a straight line: The slope of the blue straight line connecting A and B is the average acceleration in the time interval . The slope of the green line is equal to the instantaneous acceleration at point B.

f it t t

*Notice:

+ For the case of motion in a straight line, the direction of the velocity of an object and the direction of its acceleration are related as follows: When the object’ s velocity and acceleration are in the same direction, the object is speeding up. On the other hand, when the object’ s velocity and acceleration are in opposite direction, the object is slowing down.

+ From now on we shall use the term acceleration to mean instantaneous acceleration. When we mean average acceleration, we shall always use the adjective average.

2.4 Motion diagram:

2.5 One dimensional motion with constant acceleration:

-This is the case, the average acceleration over any time interval is numerically equal to the instantaneous at any instant within the interval, and the velocity changes at the same rate throughout motion.-From equation 2.6, we find that:

xa

xa

0xf xi

x

v va

t

2.9xf xi xor v v a t

- Because the acceleration is constant, we can express the average velocity in any time interval as the arithmetic mean of the initial velocity and the final velocity : xivxfv

2.102

xi xfx

v vv

- From equations 2.1, 2.2, and 2.10, we find that:

1 12.11

2 2f i xi xf f i xi xfx x vt v v t x x v v t

- Substituting equation 2.9 into equation 2.11:

21

2

12.12

2f i xi x f i xi xi xx x v v a t t x x v t a t

- Finally, substituting the value of t from equation 2.9 into equation 2.11, we can obtain:

2 2

2 2

2 2.13

1

2 2xf xi x

xf xi x

f xif i xi xf

x x

f i

v v v vx x v v

a a

v v a x x

* Note: From equation 2.9 to equation 2.13 are kinematic equations that may be used to solve any problem involving one dimensional motion at constant acceleration.

2.6 Freely falling objects:

-A freely falling object is any object moving freely under the influence of gravity alone, regardless of its initial motion.

-Objects thrown upward or downward and those released from rest are all falling freely once the are released.

-Any freely falling object experiences an acceleration directed downward, regardless of its initial motion. We shall denote the mgnitude of the three fall acceleration by the symbol g.

- If we neglect air resistance and assume that the free – fall acceleration does not vary with altitude over short vertical distances, the value of g is approximately 9,8m/s2. For making quick astimates , use .

- The motion of a freely falling object moving vertically

is equivalent to motion in one dimension under constant acceleration. Thus, the equations developed in section 2.5 for objects moving with constant acceleration can be applied.

210 /g m s

*Note:

+ The motion is in the vertical direction ( the y direction) and it is common to definde “ up” as the +y direction and free fall acceleration is downward. Thus, we always choose

+ Use y as the position variable in the kinematic equations.

210 /xa g m s

Example 2.12 Not a bad throw for a Rookie!

C the time at which the stone returns to the height from which it was thrown.D the velocity of the stone at this instant.E the velocity and position of the stone at t = 5s.

2max

2

1

21

0 20.2,04 9,8 2,042

20,4

B A xA y

B

y y y v t a t

y

m

Solution:

yB yA yv v a t

0 20 9,8

202,04

9,8B

t

t t s

2

2

1

2

0 0 20. 4,9

C A yA yy y v t a t

t t

D The value for t found in C can be inserted into equation 2.9 to give:

20 9,8 .4,08

20 /

yC yA yv v a t

m s

the velocity of the stone when it arrives back at its original height is equal in magnitude to its initial velocity but opposite in direction.

E + We can calculate this from equation 2.9, where we take t = 2,96s:

0 9,8 .2,96

29 /

yD yB yv v a t

m s

20 9,8 .5

29 /

yD yA yv v a t

m s

+ To further demonstrate that we can choose different initial instants of time, let us use equation 2.12 to find the

2

2

1

21

0 20 . 5 4,08 9,8 5 4,082

22,5

D C yC yy y v t a t

m

2.7 Kinematic equations derived from calculus:

-Mathematically, the velocity equals the derivative of the position with respect to time. It is also possible to find the position of a particle if its velocity is known as a function of time. In calculus, the procedure used to perform this task is referred to either as integration or as finding the antiderivative.

-We now use the defining equation for acceleration and velocity to derive two of our kinematic equation 2.9 and 2.12.

The definding equation for acceleration 2.7:

xx

dva

dt

May be written as , in terms of an integral ( or antiderivative), as:

x xdv a dt

0

t

xf xi xv v a dt

For the special case in which the acceleration is constant, can be removed from the integral to give

xa

0

0 2.16t

xf xi x xxv v a dt a t a t

Now let us consider the defining equation for velocity 2.5:

x

dxv

dt

We can write this as , or in integral form as xdx v dt

0

t

f i xx x v dt Because , this expression becomes:

fx x xi xv v v a t

0 0 0

2 10 0

2

t t t

f i xi x xi x

f i xi x xi

x x v a t dt v dt a t dt

tx x v t a v t

SUMMARY:

- The displacement of a particle moving along the x axis :

2.1f ix x x

- The average velocity of a particle during some time interval:

2.2x

xv

t

- The average speed of a particle :

tan2.3

total dis ceAverages speed

total time

- The instantaneous velocity of a particle:

0

lim 2.5xt

x dxv

t dt

- The instantaneous speed of a particle is equal to the magnitude of its instantaneous velocity.

- The average acceleration of a particle:

2.6xf xixx

f i

v vva

t t t

- The instantaneous acceleration of a particle:

0

lim 2.7x xx

t

v dva

t dt

- The equation of kinematics for a particle moving along the x axis with constant acceleration are:

2

2 2

2.9

12.11

21

2.122

2 2.13

xf xi x

f i x i xi xf

f i xi x

xf xi x f i

v v a t

x x v t x v v t

x x v t a t

v v a x x