physics 1d03 - lecture 41 kinematics in two dimensions position, velocity, acceleration vectors...

Physics 1D03 - Lecture 4 1

Kinematics in Two DimensionsKinematics in Two Dimensions

• Position, velocity, acceleration vectors

• Constant acceleration in 2-D

• Free fall in 2-D

Serway and Jewett : 4.1 to 4.3


The Position vector points from the origin to the particle.

r

The components of are the coordinates (x,y) of the particle:

For a moving particle, , x(t), y(t) are functions of time.

ji yxr

)(tr

r

x

y

r

(x,y)

path

xi

yj


if rrr

Displacement :

Instantaneous Velocity :

is tangent to the

path of the particle

dtrdv /

Average Velocity :

(a vector parallel to )r

tavg /rv

x

y

ir

final

initialfr

r

vavg

x

y

r

v


Acceleration is the rate of change of velocity :

)(tv

)( ttv

t time

tt time

path of particle

)(tv

)( ttv

v

tv

a

tv

t

lim

0

dtvd

a


a is the rate of change of v (Recall: a derivative gives the “rate of change” of function wrt a variable, like time).

Velocity changes ifi) speed changesii) direction changes (even at constant speed)iii) both speed and direction change

In general, acceleration is not parallel to the velocity.


Concept Quiz

A pendulum is released at (1) and swings across to (5).

143

52

a) at 3 only

b) at 1 and 5 only

c) at 1, 3, and 5

d) none of the above

0a

At which positions is ?(consider tangential a only!)


kji zyxr

(i, j, k, are unit vectors)

kji

kji

zyx vvv

dtdz

dtdy

dtdxdtrd

v

the unit vectors are constants

We get velocity components by differentiation:

Components: Each vector relation implies 3 separate relations for the 3 Cartesian components.


2

2

2

2

2

2

,

,

,

dtzd

dtdv

adtdz

v

dtyd

dt

dva

dtdy

v

dtxd

dt

dva

dtdx

v

zzz

yyy

xxx

kji

dtdv

dt

dv

dt

dv

dtvd

a zyx

Each component of the velocity vector looks like the 1-D “velocity” we saw earlier. Similarly for acceleration:


Common Notation – for time derivatives only, a dot is often used:

rvdt

vda

rdt

rdv


Constant Acceleration + Projectile Motion

a

If is constant (magnitude and direction), then:

22

1 t )(

)(

tavrtr

tavtv

oo

o

Where are the initial values at t = 0.oo vr

,

In 2-D, each vector equation is equivalent to a pair of component equations:

22

1

22

1

t)(

t)(

tavyty

tavxtx

yoyo

xoxo

Example: [down] m/s 8.9 :fall Free 2ga


Shooting the Gorilla

Tarzan has a new AK-47. George the gorilla hangs from a tree branch, and bets that Tarzan can’t hit him. Tarzan aims at George, and as soon as he shoots his gun George lets go of the branch and begins to fall.

Where should Tarzan be aiming his gun as he fires it?

A) above the gorillaB) at the gorillaC) below the gorilla


v0

r(t) =r0+v0t +(1/2)gt2

v0t

(1/2)gt2

a=g

r0


Concept quiz

Your summer job at an historical site includes firing a cannon to amuse tourists. Unfortunately, the cannon isn’t properly attached, and as the cannonball shoots forward (horizontally) the cannon slides backwards off the wall.

If the cannon hits the ground 2 seconds later, the cannonball will hit the ground:

a) 2 seconds after firing

b) 100 seconds after firing

c) seconds after firing

d) Other (explain)

1002

2 m/s 100 m/s


Example Problem

A stone is thrown upwards from the top of a 45.0 m high building with a 30º angle above the horizontal. If the initial velocity of the stone is 20.0 m/s, how long is the stone in the air, and how far from the base of the building does it land ?


Example Problem: Cannon on a slope.

How long is the cannonball in the air, and how far from the cannon does it hit?Try to do it two different ways: once using horizontal and vertical axes, once using axes tilted at 20o.

20° d

30°

100 m/s


Show that for:

2

2

cos

)sin(cos2

g

vd o

θΦ

d

vo


Summary• position vector points from origin to a particle

• velocity vector

• acceleration vector

• for constant acceleration, we can apply 1-D formulae to each component separately

• for free fall in uniform , horizontal and vertical motions are independent

r

dtrd

v

zero] to go , as , [ vttv

dtvd

a

g

physics 1d03 - lecture 41 kinematics in two dimensions position, velocity, acceleration vectors...

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