Bài giảng vật lý đại cương chương 2

Defining a Coordinate SystemOne-dimensional coordinate system consists of: • a point of reference known as the origin the origin or zero point, • a line that passes through the chosen or

Chapter 2 Motion in 1-D

2.0 Some mathematical concepts

2.1 Position, Velocity and Speed

2.2 Instantaneous Velocity and Speed

Defining a Coordinate System

One-dimensional coordinate system consists of:

• a point of reference known as the origin the origin (or zero point),

• a line that passes through the chosen origin called a

coordinate axis, one direction along the coordinate axis, chosen as positive and the other direction as negative, and the units we use to measure a quantity

Scalars and Vectors

• A scalar quantity is one that can be described with a

single number (including any units) giving its magnitude magnitude

• A Vector must be described with both magnitude and magnitude and

direction

A vector can be represented by an arrow:

•The length of the arrow represents

the magnitude (always positive) of

the vector

•The direction of the arrow represents

the direction of the vector.

A component of a vector along an axis

A one-dimensional vector can be constructed by:

•Multiply the unit vector by the magnitude of the vector

•Multiply a sign: a positive sign if the vector points to the same direction of the unit vector; a negative sign if the vector points to the opposite direction of the unit vector

A component of a vector along an axis=sign × magnitude

Difference between vectors and scalars

• The fundamental distinction between

scalars and vectors is the characteristic characteristic

of direction Vectors have it, and scalars

do not.

• Negative value of a scalar means how

much it below zero; negative component

of a vector means the direction of the

vector points to a negative direction

Check Your Understanding 1

Which of the following statements, if any,

involves a vector?

(a) I walked 2 km along the beach

(b) I walked 2 km due north along the beach.

(c) I jumped off a cliff and hit the water traveling at

2.1 Position, Velocity and Speed

• The world, and everything in it, moves.

• Kinematics: describes

motion.

• Dynamics: deals with

the causes of motion.

One-dimensional position vector

• The magnitude magnitude of the position vector is a scalar that

denotes the distance between the object and the origin

• The direction direction of the position vector is positive when the object is located to the positive side of axis from the origin and negative when the object is located to the negative side of axis from the origin

• DISPLACEMENT is defined as the change of an object's position that occurs during a period of time

• The displacement is a vector that points from an object’s

initial position to its final position final position and has a magnitude that equals the shortest distance between the two

positions

• SI Unit of Displacement: meter (m)

Example 2: Determine the displacement in the following cases:

(a) A particle moves along a line from

to

(b) A particle moves from to

(c) A particle starts at 5 m, moves to 2 m, and then returns to 5 m

EXAMPLE 3: Displacements

Three pairs of initial and final positions along

an x axis represent the location of objects

at two successive times: (pair 1) –3 m, +5 m; (pair 2) –3 m, –7 m; (pair 3) 7 m, –3 m

• (a) Which pairs give a negative

displacement?

• (b) Calculate the value of the displacement

in each case using vector notation.

Velocity and Speed

A student standing still at a

horizontal distance of 2.00

m to the left of a spot of the

sidewalk designated as the

origin.

A student is walking slowly Her horizontal position

starts at a horizontal

distance of 2.47 m to the left of a spot designated as the origin She is speeding

up for a few seconds and then slowing down

Average Velocity

• SI Unit of Average Velocity: meter per second (m/s)

i t

x t

x v

form Vectorial

t

x v

vx x

r

r r

nt

Displaceme velocity

Average

Use this form from now on

Example 4 The World’s Fastest Jet-Engine Car

Figure (a) shows that the

car first travels from left to right and covers a

distance of 1609 m in a time of 4.740 s

Figure (b) shows that in the

reverse direction, the car covers the same distance

in 4.695 s

From these data, determine the average velocity for each run.

• Example 5: find the average velocity for

the student motion represented by the

graph shown in the figure below between

the times t1 = 1.0 s and t2 = 1.5 s.

Check Your Understanding

A straight track is 1600 m in length A

runner begins at the starting line, runs due east for the full length of the track, turns

around, and runs halfway back The time for this run is five minutes What is the

runner’s average velocity, and what is his average speed?

EXAMPLE 6

You drive a truck along a straight road for 8.4 km at 70 km/h, at which point the truck runs out of gasoline and stops Over the next 30 min, you walk another 2.0 km farther along the road to a gasoline station

• (a) What is your overall displacement from the

beginning of your drive to your arrival at the station?

• (b) What is the time interval from the beginning of your drive to your arrival at the station? What is your average velocity from the beginning of your drive to your arrival

at the station? Find it both numerically and graphically Suppose that to pump the gasoline, pay for it, and walk back to the truck takes you another 45 min What is your average speed from the beginning of your drive to your return to the truck with the gasoline?

2.2 Instantaneous Velocity and Speed

The instantaneous velocity is the derivative of the object’s position with respect to time

• The instantaneous velocity of an object can be obtained by taking

the slope of a graph of the position component vs time at the point associated with that moment in time

• Instantaneous speed, which is typically called simply speed, is just

the magnitude of the instantaneous velocity vector,

Example 7

The following equations give the position component,

x(t), along the x axis of a particle's motion in four

situations (in each equation, x is in meters, t is in

2.3 Acceleration

SI Unit of Average Acceleration: meter

per second squared (m/s2)

Change in velocity Average acceleration=

Instantaneous acceleration:

22

dv d dx d x a

• An object is accelerated even only direction

changes (e.g uniformly circular motion, next

speed up or slow down.

A cat moves along an x axis What is the sign of

its acceleration if it is moving

(a) in the positive direction with increasing speed, (b) in the positive direction with decreasing speed, (c) in the negative direction with increasing speed, and

(d) in the negative direction with decreasing

speed?

EXAMPLE 7: Position and Motion

A particle's position on the x axis is given by with x in meters and t in seconds

• (a) Find the particle's velocity function and acceleration function

• (b) Is there ever a time when

• (c) Describe the particle's motion for v =x 0 ?

0

t ≥

2.4 Motion diagram

2.5 Motion with

Constant Acceleration 2.6 Free-fall

Acceleration

2.7 Kinematic

equations

Free-Fall Acceleration

• 1564 - 1642

• Applied scientific method

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

• Also looked at:

– Inclined planes – Relative motion – Thermometers – Pendulum

Equations of Motion with Constant Acceleration

21

Example 8 A Falling Stone

A stone is dropped

from rest from the

top of a tall building

Example 9 An Accelerating Spacecraft

The spacecraft shown in the

figure beside is traveling with

a velocity of +3250 m/s

Suddenly the retrorockets

are fired, and the spacecraft

begins to slow down with an

acceleration whose

magnitude is 10.0 m/s2

What is the velocity of the

spacecraft when the

displacement of the craft is

+215 km, relative to the point

where the retrorockets began

firing

Example 10

Spotting a police car, you brake your Porsche from a speed of 100 km/h to a speed of 80.0 km/h during a displacement of 88.0 m, at a

constant acceleration

• What is that acceleration?

• (b) How much time is required for the given decrease in speed?

Graphical Integration in Motion Analysis

I n t e g r a t i o n

i n

M o t i o n

Conceptual Question

1 A honeybee leaves the hive and travels 2 km before

returning Is the displacement for the trip the same as the distance traveled? If not, why not?

2 Two buses depart from Chicago, one going to New York

and one to San Francisco Each bus travels at a speed

of 30 m/s Do they have equal velocities? Explain

3 One of the following statements is incorrect (a) The car

traveled around the track at a constant velocity (b) The car traveled around the track at a constant speed

Which statement is incorrect and why?

4 At a given instant of time, a car and a truck are

traveling side by side in adjacent lanes of a highway The car has a greater velocity than the truck Does

the car necessarily have a greater acceleration?

Explain

5 The average velocity for a trip has a positive value Is

it possible for the instantaneous velocity at any point during the trip to have a negative value? Justify your answer

6 An object moving with a constant acceleration can

certainly slow down But can an object ever come to a permanent halt if its acceleration truly remains

constant? Explain