Physics 218

Physics 218

Alexei Safonov

Lecture 6: Kinematics in 2/3-D

Example Problem: Football Punt• A football is kicked at angle Q0 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

Chapter 3, continued

• Kinematics in Two or Three Dimensions–Relative Motion–Circular Motion

In the previous problem, what angle minimizes the horizontal distance

traveled?

A. q=10 degreesB. q=30 degreesC. q=60 degreesD. q=90 degrees

Vast majority don’t feel very confident about this material.

Relative and Circular MotionResults of the poll in fliptitphysics:• How familiar are you with the concepts of relative

motion and centripetal acceleration from your high school course.

A) I already know this stuffB) It seems familiar, but I need a reviewC) We didn't learn this in high school

Relative Motion

What you just did bcabac vvv

Question

x

y

vbelt,ground = 2 m/s

vdog,belt = 8 m/s

A) 6 m/s B) 8 m/s C) 10 m/s

CheckPoint• A girl stands on a moving sidewalk that moves to the right at 2 m/s

relative to the ground. A dog runs toward the girl in the opposite direction along the sidewalk at a speed of 8 m/s relative to the sidewalk.

• What is the speed of the dog relative to the ground?

vdog, ground = vdog, belt + vbelt, ground

= (-8 m/s) + (2 m/s) = -6 m/s

+x


vdog,belt = 8 m/s

What is the speed of the dog relative to the ground?

A) 6 m/s B) 8 m/s C) 10 m/s

About 55% of you got this right – lets try it again.

CheckPoint• A girl stands on a moving sidewalk that moves to the right at 2 m/s

relative to the ground. A dog runs toward the girl in the opposite direction along the sidewalk at a speed of 8 m/s relative to the sidewalk.

• What is the speed of the dog relative to the girl?


vdog,belt = 8 m/s

A) 6 m/s B) 8 m/s C) 10 m/s

A) Because the girl is actually moving and the two vectors are opposite, so together they make 6 m/s

B) Because the girl is not moving relative to the belt, and the dog is going 8 m/s relative to the belt, the dog is also moving 8 m/s relative to the girl..

C) The dog and girl are running towards each other so when you add the two velocities together it would be 8+2.

What is the speed of the dog relative to the girl?


vdog,belt = 8 m/s

A) 6 m/s B) 8 m/s C) 10 m/s

A) Because the girl is actually moving and the two vectors are opposite, so together they make 6 m/s

B) Because the girl is not moving relative to the belt, and the dog is going 8 m/s relative to the belt, the dog is also moving 8 m/s relative to the girl..

B) Because the girl is not moving relative to the belt, and the dog is going 8 m/s relative to the belt, the dog is also moving 8 m/s relative to the girl.

Using the velocity formula:vdog, girl = vdog, belt + vbelt, girl

= -8 m/s + 0 m/s = -8 m/s

What is the speed of the dog relative to the girl?


vdog,belt = 8 m/s

A) 6 m/s B) 8 m/s C) 10 m/s

Clicker Question

Moving sidewalk

2 m/s1 m/s

4 m

• A man starts to walk along the dotted line painted on a moving sidewalk toward a fire hydrant that is directly across from him. The width of the walkway is 4 m, and it is moving at 2 m/s relative to the fire-hydrant. If his walking speed is 1 m/s, how far away will he be from the hydrant when he reaches the other side?

A) 2 m

B) 4 m

C) 6 m

D) 8 m

Time to get across:

Dt = distance / speed = 4m / 1m/s = 4 s

1 m/s

4 m

If the sidewalk wasn’t moving:

sidewalkmanv ,

4 m

2 m/s

Just the sidewalk:

hydrantsidewalkv ,

1 m/s

4 m

2 m/s

Combination of motions:

hydrantsidewalksidwalkmanhydrantsidewalk vvv ,,,

D = (speed of sidewalk) ∙ (time to get across)= (2 m/s) ∙ (4 s) = 8 m

1 m/s

4 m2 m/s

D

• Three swimmers can swim equally fast relative to the water. They have a race to see who can swim across a river in the least time. Relative to the water, Beth swims perpendicular to the flow, Ann swims upstream at 30 degrees, and Carly swims downstream at 30 degrees.

• Who gets across the river first?

• A) Ann B) Beth C) Carly

x

y

AnnBeth

Carly

Clicker Question

A

B

C

Vy,Beth = Vo

30o 30o

Vy,Ann = Vocos(30o)

Vy,Carly = Vocos(30o)

Time to get across = D / Vy

D

Look at just water & swimmers

x

y

x

y

Clicker Question• Three swimmers can swim equally fast relative

to the water. They have a race to see who can swim across a river in the least time. Relative to the water, Beth swims perpendicular to the flow, Ann swims upstream at 30 degrees, and Carly swims downstream at 30 degrees.

• Who gets across the river second?

• A) Ann B) Carly C) Both same

Ann Carly

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 q you should aim at to do that?

q

vB

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

any values for vB and vC?

A. Yes, always possibleB. Only possible if vB>vC

C. Only possible if vB>2vC

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

q

vB

vC

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

A girl twirls a rock on the end of a string around in a horizontal circle above her head as shown from above in the diagram.

If the string breaks at the instant shown, which of the arrows best represents the resulting path of the rock?

A

B

C

D

Top view looking down

CheckPoint

A girl twirls a rock on the end of a string around in a horizontal circle above her head as shown from above in the diagram.

If the string breaks at the instant shown, which of the arrows best represents the resulting path of the rock?

A

B

C

D

Top view looking down

After the string breaks, the rock will have no force acting on it, so it cannot accelerate. Therefore, it will maintain its velocity at the time of the break in the string, which is directed tangent to the circle.

CheckPoint

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

-

aR


• Need to know a bit more math to calculate the magnitude, so for now just memorize:

• Centripetal acceleration: |a|=v2/R

dtvvdtvda /)(/ 12

-

aR



v = wR

Once around:

w Dq / Dt = 2p / T

v Dx / Dt = 2pR / T

w is the rate at which the angle q changes:

q

dtdqw

Speed = distance/timeDistance in 1 revolution divided by the

time it takes to go around onceSpeed = 2pr/T

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

Circular Motion: Get the speed!

Why do we usually neglect the acceleration due to Earth's rotation? Is it really that small? Let’s check

R=3,959 miles T=24 hrs

Vequator= 2p R/T

Another Example Problem: Rescue Plane

• You are the pilot of a rescue plane. Your mission is to drop supplies to isolated mountain climbers on a rocky ridge a height h below. If your plane is traveling horizontally with a speed of VO:

• How far in advance of the recipients (horizontal distance) must the goods be dropped?

Integration Problem• An object moves with velocity v (a=1m/s2,

b=2m/s3, v0=3m/s)

• Draw a graph of v vs t • Draw a graph of x vs t?

2btatvv o

– Ok, assume that x=5 at t=0

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Physics 218