University Physics 1: Position, Displacement, Velocity, and Acceleration, Lecture notes of Physics

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Part 2: Kinematics

Chapter 2 Motion in One Dimension

Objectives:

• To study motion along a straight line
• To define and differentiate average and instantaneous linear velocity; average and instantaneous linear acceleration
• To explore applications of straight-line motion with constant acceleration
• To examine freely falling bodies
• To consider straight-line motion with varying acceleration
• To learn some simple calculus

Position, Velocity and Speed

Position: is the object’s location with respect to a chosen reference point.

Position-time graph shows the motion of the particle (car).

The smooth curve is a guess as to what happened between the data points.

Distance is the length of a path followed by a particle. Displacement: change in position during some time interval. ƒ Represented as ' x ; ' x ≡ x (^) f - x (^) i ƒ SI unit: meter (m) ƒ ' x can be positive or negative

Example: A player moves from one end of the court to the other and back.

Distance is twice the length of the court and is always positive

Displacement is zero since x (^) f = x (^) i

distance is a scalar displacement is a vector

Average speed:

Average velocity:

Position, Displacement and Average Velocity

t

x t t

x x v f i

f i '

t

d v '

Example: Find the displacement, average velocity and average speed of the car between positions A and F.

Ans:

Displacement

Average velocity

Average speed

'x xF xA  53  30  83 m

1. 7 m/s 50

t

x v (^) xavg

2. 5 m/s 50

't

d vavg

• Instantaneous speed is the magnitude of the instantaneous velocity.
• Average speed, in general, is unequal to the magnitude of average velocity
• The instantaneous speed has no direction associated with it.

Vocabulary Note

“Velocity” and “speed” will indicate instantaneous values.

Average will be used when the average velocity or average speed is indicated.

Summary on derivatives ( 导数，微商 )

Function y=f (x) Derivative y c =f c (x)=df/dx

a 0 x n^ nx n- sin(ax) acos(ax) cos(ax) - asin(ax) tan(ax) asec 2 (ax) ln(ax) 1/x e ax^ ae x ax^ axlna

Basic rules for differentiation ( 求导数，微分法 )

Product Rule: (^) > @ f x dx

g x g x^ d dx

f x g x f x^ d dx

d (^) 

@

x x x x x x x

x dx

x x^ d dx

x x x^ d dx

d

2 1 9 3 4 4 30 9 16

2 1 3 4 2 1 3 4 3 4 2 1 2 2 3 4 2

2 3 2 3 3 2

    

Example:       

Quotient Rule:

g x @^2

g x f x f x g x g x

f x dx

d c  c » ¼

º « ¬

ª

@^2

1 g x

g x dx g x

d (^)  c » ¼

º « ¬

ª

Example: 2 3 4 2

8 3 4

3 4 2 2 3 4 3 4

2  

   ¸ ¹

¨^ · ©

§  (^) x x

x dx

x x^ d dx

x^ d x

x dx

d

Basic rules for differentiation ( 求导数，微分法 )

Chain Rule: (^) g x dx

f g x^ d dg x

f g x^ d dx

d

Example: Differentiate cos(2 x +3). Let g ( x )=2 x +3 and f (g)=cos g

Example: Differentiate

f g x fc^ g gc x sin g x ˜ 2  2 sin 2 x 3 dx

d

x^2  1

1

2 2

1 c c ˜ x (^2)  x^ x g

f g x f g g x dx

d

General Power Rule: (^) > @ > @ f x dx

f x r f x^ d dx

d (^) r r 1

x (^) exp 2 x^2 @ (^) exp 2 x^2 x>exp (^2) x^24 x @ (^14) x^2 exp 2 x^2 dx

d (^)       

Example:

Acceleration ( 加速度 )

t

v t t

v v a f i

f i '

Instantaneous acceleration: 2

2 lim (^0) dt

d x dt

dx dt

d dt

dv t

v a (^) t ¸ ¹

' o

Acceleration is the rate of change of the velocity.

Dimensions are L/T^2 ; SI units are m/s²

Graphical Comparison

• Given the x-t graph (a)
• v-t graph: the slope of the x-t graph at every instant.
• a-t graph: the slope of the v-t graph at every instant.

Acceleration and Velocity, Directions

• When an object’s v and a are in the same direction, the object is speeding up.
• When an object’s v and a are in the opposite direction, the object is slowing down.

Motion Diagrams A snapshot of a moving object at equal time intervals.

Images are equally spaced. Acceleration equals zero.

Images become farther apart as time increases. Velocity and acceleration are in the same direction. Acceleration is uniform. Velocity is increasing.

Images become closer together as time increases. Acceleration and velocity are in opposite directions. Acceleration is uniform. Velocity is decreasing.

Kinematic Equations under Constant Acceleration Kinematic equations can be used with any particle under uniform acceleration.

x avg ,^ xi^ 2 xf v v^  v

When the acceleration is zero, the constant acceleration model reduces to the constant velocity model.