Í kinematics in two dimensions

4. Motion in 2

4. Motion in 2 - - & 3 & 3 - - Dimensions Dimensions

z

Chapter 4

Í kinematics in two dimensions

» projectile motion Í circular motion

Í relative motion

Summary of Ch. 2 (Motion along a straight line)

• equations with constant acceleration

z

Δx = v

t + 1/2 at

z

Δv = at

z

v

= v

+ 2a Δx

z

y = y

+ v

t - 1/2 gt

z

v

= v

- gt

z

v

= v

- 2gΔy

• free-fall: a

= -g = -9.81 m/s

4 4 - - 2 Position and Displacement 2 Position and Displacement

Average velocity

z Position Vector : r = xi + yj +zk

z Displacement : Δr = r

-r

=Δxi + Δyj +Δzk

4 4 - - 3 Average velocity and 3 Average velocity and Instantaneous velocity Instantaneous velocity

4 4 - - 3 Average acceleration and 3 Average acceleration and Instantaneous acceleration Instantaneous acceleration

Average acceleration

™ Scalar components of a(t) :

(Sample problems 4

(Sample problems 4- -2, 3) 2, 3)

$

2 2

( ) 0.31 7.2 28 ( ) 0.22 9.1 30 ( ) ( ) ( )

x t t t

y t t t

r t x t i y t j

= − + +

= − +

= +

r $

( ) At a t =15 ,s

( ) 15

=

속도

( 0.312 7.2 28) 0.62 7.2

x

dx d

v t t

dt dt t

= = − + +

= − +

(순간)속도는 접선방향

(b) t=0~15 s 경로

Kinematics in Two Dimensions Kinematics in Two Dimensions

z

x = x

+ v

t + 1/2 a

t

z

v

= v

+ a

t

z

v

= v

+ 2a

Δx

z

y = y

+ v

t + 1/2 a

t

z

v

= v

+ a

t

z

v

= v

+ 2a

Δy

x and y motions are independent!

They share a common time t

4 4 - - 5 Projectile motion 5 Projectile motion

a a

= 0 a = 0 a

= - = - g g

Kinematics for Projectile Motion Kinematics for Projectile Motion

z

x = x

+ v

t

z

v = v

z

y = y

+ v

t - 1/2 gt

z

v

= v

gt

z

v

= v

- 2g Δy

Again,

x and y motions are independent!

They share a common time t

a

= 0 a

= -g

4- 4 -6 Projectile motion analyzed 6 Projectile motion analyzed

Projectile equation h

when θ=90^o

4- 4 -7 Uniform circular motion 7 Uniform circular motion

질질 문문

1) 등속 원운동은 등속도 운동이다? 예, 아니오

2) 등속 원운동은 등가속도 운동이다? 예, 아니오

http://www.mcasco.com/

Report : proof of Eq. 4-34

Rotating

during

Angular velocity

Angular velocity : Angular velocity :

( )

2 1 , : radius

0 ,

r r r r r

r d r d d

r t r

t t dt dt dt

v r θ

θ θ ω θ

ω Δ = − = Δ

Δ = Δ ⇒ Δ → ⇒ = ≡

Δ Δ

∴ = uur ur ur uuur

uur uuur r r r

r ur

v r = r ω ur

v ω = r

ur r

ω=2π

Period T

1

2 2

T f

r v π ω

π

=

=

=

Rotating

during

Acceleration (radial) Acceleration (radial)

v

Δθ a

( )

2 1

2 2

, : speed

0

v v v v v

v dv d

v t a v v

t t dt dt

a a v r v

r θ

θ θ ω

ω ω Δ = − = Δ

Δ = Δ ⇒ Δ → ⇒ = = =

Δ Δ

= = = =

uur uur ur uuur

uur uuur r r

r ur

r

2 1

, : speed

v v v v θ v

Δ = − = Δ

uur uur ur uuur

Tangential and Radial Acceleration

a r

: Rotational Motion

a r

: Acceleration along the tangential direction

r a

v

=

a r

a r

a r

Relative velocity Relative velocity

v _g = v _a + u

4 4 - - 8,9 Relative motion 8,9 Relative motion

.) (

v cons a

a

v v

v

r r

r

BA PB

PA

BA PB

PA

=

=

+

=

+

=

Q

Shooting the Monkey...

Shooting the Monkey...

y = vy v_0y t - ¹/₂ g tg ²

z

Still works even if you shoot upwards!

Dart hits the

monkey!

Shooting the monkey...

Shooting the monkey...

x = x x = x₀₀

y = -y ¹/₂ g tg ² xx = v= v_{0 0} tt

y = -y ¹/₂ g tg ²

Question Question

A flatbed railroad car is moving along a track at constant velocity. A

passenger at the center of the car throws a ball straight up. Neglecting air resistance, where will the ball land?

1. Forward of the center of the car 2. At the center of the car

3. Backward of the center of the car

correct

The ball has no acceleration in the horizontal direction. Therefore, the balls remains directly above the center of the train at all times during the flight and would fall directly back toward the center of the train.

Great Answers!

The train and the ball have the same horizontal velocity and by throwing the

ball straight up, the horizontal component is not changed.

Question Question

You and a friend are standing on level ground, each holding identical baseballs. At exactly the same time, and from the same height, you drop your baseball without throwing it while your friend throws her baseball horizontally as hard as she can. Which ball hits the ground first?

1. Your ball

2. Your friends ball

3. They both hit the ground at the same time correct

Question (great answers) Question (great answers)

They both have the same initial vertical component with the same acceleration due to gravity, therefore they hit the ground at the same time.

No matter how much horizontal velocity is put on an object it still falls at the same rate as any other dropped object.

• y = y

+ v

t - gt

/2

• v

= 0 and y=0

• therefore t=sqrt(2y

/g)

• result is independent of v

Question Question

(common misconceptions) (common misconceptions)

My ball has less distance to travel.

The friend's ball is actually thrown, while your ball is only dropped without any extra force. Your friend's ball has more inertia, so

therefore it will reach the ground quicker and with much more force.

Friend's ball must travel horizontally then downward. My ball just travels downward.

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. On the other hand, you also know that the dart will not travel in a straight line, but rather in a parabolic path like any other projectile. In order to hit the monkey with the dart, where should you point the gun before shooting?

1 Right at the monkey 2 Below the monkey 3 Above the monkey

Question Question

correct

If the shot is fired at the monkey the same time the monkey drops, both objects will fall at the same rate causing the shot to hit the monkey.

since the monkey is going to start falling right away you need to aim below it

Along the way, gravity is going to pull the dart

down, so you would have to aim up. Aiming right at it or below would miss the monkey by going

underneath it.

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 (B) swims perpendicular to the flow, Ann (A) swims upstream, and Carly (C) swims downstream. Which swimmer wins the race?

A) Ann B) Beth C) Carly

Question Question

correct

A B C

Question Question

(great answers) (great answers)

z

Beth will reach the shore first because the vertical component of her velocity is greater than that of the other swimmers.

z

The key here is how fast the vector in the vertical

direction is. "B" focuses all of its speed on the vertical vector, while the others divert some of their speed to the horizontal vectors.

Time to get across =

width of river/vertical component of velocity

A B C

Question Question

(common misconceptions) (common misconceptions)

• While Carly is moving forward she will also be moving along with the current. two positive(+) direction motions = faster velocity.

• Carly will get there first because she is using the current to her advantage.

A B C

Follow

Follow- -up Question up Question

Heather wants to swim across a flowing river in such a way that she ends up on the opposite side directly opposite her starting point. She should therefore aim….

1) upstream 2) downstream 3) directly across

v

v

v

^v

^v

^v

r r

r = +

correct

Summary

1. Projectile Motion

2. Relative Motion

z

x = x

+ v

t

z

v = v

z

y = y

+ v

t - 1/2 gt

z

v

= v

gt

z

v

= v

- 2g Δy

sw

sg

v v

v r = r + r