Physics 218

Alexei Safonov

Lecture 5: Kinematics in 2-3D

Checklist for Today

•  For the rest of this week: –  Be ready for the lab (WebAssign) including pre-labs

•  Did you get my email about the question on the uncertainty for volume in problem #4?

–  Quiz at the recitation on Chapter 2 –  There was a pre-lecture for today

•  Many people completed, but not all

–  A pre-lecture due Wednesday morning •  Don’t forget the checkpoints

–  Homework for Chapter 2 is due Sunday on MP •  Coming week recitation quiz will be on Chapter 3

Speeder A speeder passes you (a police officer) sitting by the side of the road and maintains their constant velocity V. You immediately start to move after the speeder from rest with constant acceleration a. • How much time does it take to ram the speeder? • How far do you have to travel to catch the speeder? • What is your final speed?

X Police Officer Speeder

Throw a Ball up You throw a ball upward into the air with

initial velocity V0. Calculate: a)  The time it takes to reach its highest

point (the top). b)  Distance from your hand to the top c)  Time to go from your hand and come

back to your hand d)  Velocity when it reaches your hand e)  Time from leaving your hand to reach

some random height h.

Chapter 3

•  Kinematics in Two or Three Dimensions •  Projectile Motion •  Uniform Circular Motion

Important Equations of Motion If the acceleration is constant

Position, velocity and Acceleration are vectors.

221

00

0

tatv x x

tav v

++=

+=

Projectile Motion

The horizontal and vertical equations of the motion behave

independently Problem solving:

The trick for all these problems is to break them up into the X and Y directions.

Constant Acceleration

2y2

10y0

2x2

10x0

tat v y y

tat v x x

++=

++=

t a v v tat v r R

0

221

00

+=

++=

ta v vta v v

y0yy

x0xx

+=

+=

Kinematics in 3D

Projectile Motion & Frames of Reference

Checkpoint 1

A)  The  launched  marble  hits  first.  B)  The  dropped  marble  hits  first.  C)  They  both  hit  at  the  same  :me.    

Lets  check  again  

A  physics  demo  launches  one  marble  horizontally  while  at  the  same  instant  dropping  a  second  marble  straight  down.  Which  one  hits  the  ground  first?    

Ball Dropping •  Analyze Vertical

and Horizontal separately!!!

•  Ay = g (downwards) •  Ax = 0

– Constant for Both cases!!!

Vx = 0 Vx>0

Projectile Motion

The horizontal and vertical equations of the motion behave

independently Problem solving:

The trick for all these problems is to break them up into the X and Y directions.

Projectile Motion Horizontal Vertical Boring

Monkey Troubles • You are a vet trying to shoot a tranquilizer dart into a monkey hanging from a branch in a distant tree. You know that the monkey is very nervous, and will let go of the branch and start to fall as soon as your gun goes off. In order to hit the monkey with the dart, where should you point the gun before shooting?

• A) Right at the monkey • B) Below the monkey • C) Above the monkey

Monkey x = xo

Dart  x = vo t

2

21 gty −=

Shooting the Monkey…

2

21 gty −=

Shooting the Monkey…

y = voy t - 1/2 g t 2

S:ll  works  even  if  you  shoot  upwards!  y = yo - 1/2 g t 2

Dart  hits  the  monkey    

Checkpoint 2 •  A destroyer simultaneously fires two shells with the same initial

speed at two different enemy ships. The shells follow the trajectories shown. Which ship gets hit first.

Enemy  1  Destroyer   Enemy  2  

A)  Enemy  1  B)  Enemy  2  C)  They  are  both  hit  at  the  same  :me    

Checkpoint 2 •  …Which enemy ship gets hit first? •  A) Enemy 1 B) Enemy 2 C) Same

B)  Both  shots  were  accelera:ng  towards  the  ground  at  the  same  rate,  but  the  shot  fired  at  Enemy  2  did  not  go  as  high  and  therefore  took  less  :me  to  fall  back  to  the  ground.    

C)  we  are  given  that  two  shells  are  fired  at  same  speed.  therefore,  both  ships  should  get  hit  at  the  same  :me.    

A)  they  are  traveling  at  the  same  speed,  but  the  enemy  one  trajectory  is  shorter    

Enemy  1  Destroyer   Enemy  2  Enemy  1   Enemy  2  

Enemy  1  Destroyer  

Checkpoint 3 •  A destroyer fires two shells with different initial speeds at two

different enemy ships. The shells follow the trajectories shown. Which enemy ship gets hit first?

Enemy  2  

A) Enemy 1 B) Enemy 2 C) They are both hit at the same time

Enemy  1  Destroyer   Enemy  2  

B)  The  ini:al  speed  of  the  shell  fired  at  ship  2  is  greater,  so  ship  2  gets  hit  first.    

C)  they  both  achieve  the  same  height  so  they  remain  in  the  air  the  same  amount  of  :me    

A)  Since  the  ini:al  speed  for  the  first  shell  is  lower,  Enemy  Ship  1  will  be  hit  first.  

•  …Which enemy ship gets hit first? •  A) Enemy 1 B) Enemy 2 C) Same

Checkpoint 3

Marbles & Math

•  Prove mathematically that an object projected horizontally will reach the ground at the same time as an object dropped vertically

FINISHED HERE

Firing up in the air at an angle A ball is fired up in the air with velocity Vo and angle Θo. Ignore air friction. The acceleration due to gravity is g pointing down. What is the final velocity here?

Football Punt •  A football is kicked at angle Θ0 with a

velocity V0. The ball leaves the punters foot h meters above the ground. – The velocity at the maximum height – How far does it travel, in the X direction, before

it hits the ground? – What angle maximizes the distance traveled

h

In the previous problem, which of the given angles minimizes the horizontal distance traveled?

A.  θ=10 degrees B.  θ=30 degrees C.  θ=60 degrees D.  θ=90 degrees

Checklist for Today

•  For the rest of this week: –  Be ready for the lab (WebAssign) including pre-labs

–  Quiz at the recitation on Chapter 3 –  There was a pre-lecture for today

•  Many people completed, but not all

–  A pre-lecture due Wednesday morning •  Don’t forget the checkpoints

–  Homework for Chapter 3 is due Sunday on MP •  Coming week recitation quiz will be on Chapter 3

Uniform Circular Motion

•  Fancy words for moving in a circle with constant speed

•  We see this around us all the time – Moon around the earth – Earth around the sun – Merry-go-rounds

Uniform Circular Motion - Velocity

• Velocity vector = |V| tangent to the circle

•  Is this ball accelerating? – Why?

Centripetal Acceleration

•  Vector difference V2 - V1 gives the direction of acceleration a

dtvvdtvda /)(/ 12

−≈=

a

R

Centripetal Acceleration •  “Center Seeking” •  Accel vector= V2/R

towards the center •  Acceleration is

perpendicular to velocity

)ˆ( rRv a

2

−=

R direction ̂r

Speed = distance/time Distance in 1 revolution divided by

the time it takes to go around once Speed = 2πr/T

Note: The time to go around once is known as the Period, or T

Circular Motion: Get the speed!

Ball on a String

•  A ball at the end of a string is revolving uniformly in a horizontal circle (ignore gravity) of radius R. The ball makes N revolutions in a time t.

•  What is the centripetal acceleration?

In previous problem, how would acceleration change if the number of revolutions N were to double?

A.  Acceleration will be ½ of the original value

B.  Acceleration with be double original value

C.  Acceleration will be quadruple original value

D.  Acceleration will be ¼ of the original value

Firing up in the air at an angle A ball is fired up in the air with velocity Vo and angle Θo. Ignore air friction. The acceleration due to gravity is g pointing down. What is the final velocity here?

Boat on the River •  You want to cross the

river so that the boat gets exactly from A to B. The river has a current vC=4 km/h. Your boat’s speed in still water is vB=20km/h?

•  What is the angle θ you should aim at to do that?

θ

vB

In previous problem, is it possible to get from A to B for any values for

vB and vC? A.  Yes, always possible B.  Only possible if vB>vC

C.  Only possible if vB>2vC D.  Only possible if vB>>vC (much larger)

vC

θ

vB

Next time… •  Reading: Finish Chapter 3 if you haven’t

already •  Homework:

– HW1 was due yesterday – HW2 covered in recitation this week; due

Monday (6 days from now) – Start working on HW3

•  Next time: More on kinematics in two dimensions – Reading Quiz hints: Q3.13-Q3.16

A pendulum swings in an arc, at what point (of the three points A, B, C) is the magnitude of the acceleration in the x

direction greatest? A.  A B.  B C.  C D.  Same at all points

After leaving the gun a projectile moves in a parabolic path without air

resistance. Which statement is TRUE?

A.  a is parallel to v along its path. B.  a is perpendicular to v at top of the

parabola. C.  a is perpendicular to v along its path. D.  a is parallel to v at the top of the

parabola.