Physics 1D03 - Lecture 2 1

Kinematics in One DimensionKinematics in One Dimension

• Displacement, velocity, acceleration, free fall• Examples

Knight: Chapters 1, 2

Physics 1D03 - Lecture 2 2

1-D motion can be described by scalars (real numbers with units) as functions of time:

Position x(t) (displacement from the origin)

Velocity v(t)=dx/dt (rate of change of position)

Acceleration a(t)=dv/dt (rate of change of velocity)

Physics 1D03 - Lecture 2 3

2 2

1)(

)(

constant

tatvtx

atvtv

a

o

o

advv 221

22

Caution: These assume acceleration is constant.

Using the definitions we can derive

A Special Case: Constant Acceleration

dtdx

vdtdv

a ,

From the above you can get:

Physics 1D03 - Lecture 2 4

All objects in free fall move with constant downward acceleration:

This was demonstrated by Galileo around 1600 A.D.

The constant “g” is called the “acceleration due to gravity”.

Example: Free Fall.

(“Free fall” means the only force is gravity; the motion can be in any direction).

][downwards s/m 80.9 2ga

Physics 1D03 - Lecture 2 5

Quix 1

A block is dropped from rest. It takes a time t1 to fall the first third of the distance. How long does it take to fall the entire distance?

a) t1

b) 3t1

c) 9t1

d) None of the above

3

Physics 1D03 - Lecture 2 6

Example 1

A particle’s position is given by the function:

x(t)=(-t3+4t) m

a) what is the velocity at t=3 s ?

b) what is the acceleration at 3 s ?

c) make a sketch of the motion

Physics 1D03 - Lecture 2 7

An object if thrown straight up with a velocity of 5m/s. What will the velocity be when it comes back to its original position ?

Example 2

Physics 1D03 - Lecture 2 8

Example 3 A skier is moving at 40m/s at the top of a

hill. His velocity changes to 10m/s after covering a distance of 600m. What is his acceleration ?

Physics 1D03 - Lecture 2 9

Example 3b

The skier’s girlfriend is also traveling at 40m/s, but, unfortunately, after only 3s, hits a tree and her velocity ‘suddenly’ comes to 0m/s.

How far did she get, given the same deceleration as in the previous question?

Physics 1D03 - Lecture 2 10

Vector Review

• Scalars and Vectors

• Vector Components and Arithmetic

Physics 1D03 - Lecture 2 11

Physical quantities are classified as scalars, vectors, etc.

Scalar : described by a real number with units

examples: mass, charge, energy . . .

Vector : described by a scalar (its magnitude) and a direction in space

examples: displacement, velocity, force . . .

Vectors have direction, and obey different rules of arithmetic.

Physics 1D03 - Lecture 2 12

Notation

• Scalars : ordinary or italic font (m, q, t . . .)

• Vectors : - Boldface font (v, a, F . . .)

- arrow notation

- underline (v, a, F . . .)

• Pay attention to notation :

“constant v” and “constant v” mean different things!

.) . . F ,a ,v(

Physics 1D03 - Lecture 2 13

Coordinate Systems

In 2-D : describe a location in a plane

• by polar coordinates :

distance r and angle

• by Cartesian coordinates :

distances x, y, parallel to axes with: x=rcosθ y=rsinθ

These are the x and y components of r

x

y

r

( x , y )

0 x

y

Physics 1D03 - Lecture 2 14

Addition:

Ax

Ay

A

By

Bx

B

By

Bx

B

Ay

Ax

A

C

Cx

Cy

If A + B = C ,

then:

zzz

yyy

xxx

BAC

BAC

BAC

Three scalar equations from one vector equation!

Tail to Head