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Department of Physics and Applied Physics 95.141, Fall 2013, Lecture 5

Course website: http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Lecture Capture:

http://echo360.uml.edu/danylov2013/physics1fall.html

Lecture 5

Kinematics in two dimensions

09.18.2013 Physics I

http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

http://echo360.uml.edu/danylov2013/physics1fall.html


Chapter 3: Sections 3.6 – 3.8 Vector kinematics Projectile motion

Outline


Vector Kinematics

• Kinematics in more than one dimension • Previously described 1D displacement as Δx, where

motion could only be positive or negative. • In more than 1D, displacement is a vector

v


Displacement

)1(12 Dxxx −=∆

12 rrr −=∆x

y

displacement (in unit vector notation): jyyixxr ˆ)(ˆ)( 1212 −+−=∆

1t

2tIn two or three dimensions, the displacement is a vector:

ij

jyixrjyixr ˆˆˆˆ222111 +=+=


Average Velocity

x

y

tr

ttrrv

∆∆

=−−

=

12

12

1t2t

Average velocity is the displacement divided by the elapsed time

As Δt and Δr become smaller and smaller, the average velocity approaches the instantaneous velocity.


Instantaneous velocity

1vThe instantaneous velocity indicates how fast the object moves and the direction of motion at each instant of time

2r

3r

2v

3v

x

y


Instantaneous acceleration

The instantaneous acceleration is in the direction of ,and is given by:

12 vvv −=∆

1v

2r

2v

x

y

1v

2v v∆

Average acceleration

2

2

dtrda

=


Using unit vectors

kvjvivkdtdxj

dtdyi

dtdx

dtrdv zyx

ˆˆˆˆˆˆ ++=++==

kdt

dvjdt

dvi

dtdva zyx ˆˆˆ ++=


Example Problem

Given position as a function of time, find instantaneous velocity and instantaneous acceleration at t=3s

kita ˆ2ˆ8)( +=

kikistv ˆ6ˆ24ˆ)32(ˆ)38()3( +=⋅+⋅==

Acceleration is constant


Projectile motion

A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola.


• X and Y motions are completely independent • Work the problem as two one-dimensional problems Each dimension obey different equations

Projectile motion


By splitting the equations of motion into component form, we can solve problems one direction at a time

2D Motion with constant acceleration

tavvtatvrr

+=

++=

0

221

00

There is only one parameter, which connects X and Y motion: time


)(2

02

02

221

00

0

xxavv

tatvxxtavv

xxx

xx

xxx

−+=

++=

+=

No forces in x direction.

Air resistance is neglected

Projectile motion X direction

ax = 0 2

02

xx vv =

0

0

0 xx vv 0=

00 tvxx x+=

vx is constant!!!!


Projectile motion Y- direction )(2

02

02

221

00

0

yyavv

tatvyy

tavv

yyy

yy

yyy

−+=

++=

+=

y

g

y

g

ay = -g

ay = g

if then

if then

)(2

02

02

221

00

0

yygvv

gttvyy

gtvv

yy

y

yy

−−=

−+=

−=

)(2

02

02

221

00

0

yygvv

gttvyy

gtvv

yy

y

yy

−+=

++=

+=

x

x


Projectile Motion: Concept Test 1

A helicopter moving at a constant horizontal velocity to the right drops a package when at position A. Which of the marked trajectories is closest to that observed by a stationary person on the ground?

observer


• Draw diagram, choose coordinate system

• Knowns and unknowns • Divide equations into x and y • Solve, noting that in the x and y

calculations the common parameter is the time interval t

Helicopter flying horizontally at 70m/s wants to drop supplies on mountain top 200m below. How far in advance (horizontal distance) should the package be dropped?

Example (Rescue Helicopter)

g


2

21 gttvyy oyo −+=

common parameter is time t Solve y-equations to find t Plug t into x-equations to

find x

Example (Rescue Helicopter)

mmx

447)39.6(70

==

00 =x 00 =y m/s 700 =xv 00 =yv

0=xa gay −=

?=x m 200−=y )(2

02

02

221

00

0

yygvv

gttvyy

gtvv

yy

y

yy

−−=

−+=

−=

xx

x

vvtvxx

0

00

=+=

0 0

gyt 2

−=

tvxx x00 +=

gyvx x

20 −=

0

ssm

mt 39.6/8.9

)200(22 =

−−

=

ConcepTest 2 Dropping the Ball I

From the same height (and at the same time), one ball is dropped and another ball is fired horizontally. Which one will hit the ground first?

A) the “dropped” ball B) the “fired” ball C) they both hit at the same time D) it depends on how hard the ball

was fired E) it depends on the initial height

From the same height (and at the same time), one ball is dropped and another ball is fired horizontally. Which one will hit the ground first?

A) the “dropped” ball B) the “fired” ball C) they both hit at the same time D) it depends on how hard the ball

was fired E) it depends on the initial height

Both of the balls are falling vertically under the influence of

gravity. They both fall from the same height. Therefore, they will

hit the ground at the same time. The fact that one is moving

horizontally is irrelevant—remember that the x and y motions are

completely independent !!

ConcepTest 2 Dropping the Ball I


A golf ball is hit with initial velocity v0 at an angle θ0 above the horizontal. Draw diagram and choose coordinate system

Fill in knowns

x

y

v0

v0x

v0y θ0

Example (Golf Ball)

00 =x 00 =y

0=xa gay −=000 cos θvv x = 000 sin θvv y =

g


A golf ball is hit with initial velocity v0 at an angle θ0 above the horizontal.

Find: the time of flight (how long the ball is in the air) This depends only on the y-component equations, as the motion in y direction stops the flight.

Since both y0 and y are zero at the beginning/end

y=0 when the ball was hit

The second is the time of flight gv

t y021

2 tand 0 ==

Cont. Example (Golf Ball).

0 0

y=0 when the ball landed


gvR oo θθ cossin2 0

2

=


Find: Range (how far does ball travel on flat ground) Use constant x-velocity to calculate how far ball travels

horizontally during time of flight (Range)

gv

gv

t y θsin22 002 ==time of flight

000 cos θvvv xx ==Constant x velocity

Range

Maximum Range

Cont. Example (Golf Ball)

200 tvxxR x ⋅+==

°== 4512sin 00 θθ or

gvo 0

2 2sin θ=

when

Range

0



Find: trajectory (height y as a function of position x)

Since common parameter is time t, eliminate t to get y(x)

Cont. Example (Golf Ball)

Equation of parabola

Which of the

three punts

has the

longest hang

time? D) all have the same hang time

A B C

h

The time in the air is determined by the vertical motion!

Because all of the punts reach the same height, they all

stay in the air for the same time.

ConcepTest 3. Punts I

)(2 02

02 yygvv yy −−= 2

21 tatvyy yoyo ++=

A B

C) both at the same time

A battleship simultaneously fires two shells at two enemy

submarines. The shells are launched with the same initial

velocity. If the shells follow the trajectories shown, which

submarine gets hit first ?

The flight time is fixed by the motion in the y-direction. The higher an object goes, the longer it stays in flight. The shell hitting submarine #2 goes less high, therefore it stays in flight for less time than the other shell. Thus, submarine #2 is hit first.

ConcepTest 4 Punts II


Relative Motion: Clicker Quiz

A helicopter moving at a constant horizontal velocity to the right drops a package when at position A. Which of the marked trajectories is closest to that observed by a person on the helicopter?

observer


Solving Problems Involving Projectile Motion

1. Read the problem carefully, and choose the object(s) you are going to analyze.

2. Draw a diagram.

3. Choose an origin and a coordinate system.

4. Decide on the time interval; this is the same in both directions, and includes only the time the object is moving with constant acceleration g.

5. Examine the x and y motions separately.

6. List known and unknown quantities. Remember that vx never changes, and that vy = 0 at the highest point.

7. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them.


Thank you See you on Monday

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