General Physics 1 (Phys 110 : Mechanics)

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General Physics 1

(Phys 110 : Mechanics)

CHAPTER 4

Motion in 2D and 3D

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Phys 110

1. Position vector (𝒓):

Chapter 4 : Motion in 2D and 3D

Revision :

Lesson 4 of 5 Slide 1

𝒓 𝒕 = 𝒙 𝒕 𝒊 + 𝒚 𝒕 𝒋 + 𝒛(𝒕)𝒌

Particle’s motion in

2D

Position vector 1

Position vector 2

2. Displacement vector (∆𝒓):

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Phys 110

3. Average Velocity (𝒗

_𝒂𝒗𝒈

):

Revision :

4. Instantaneous Velocity (Velocity) (𝒗):

direction of 𝒗

_𝒂𝒗𝒈

= direction of ∆𝒓

direction of 𝒗 : along

the tangent to path

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Phys 110

5. Average Acceleration (𝒂

_𝒂𝒗𝒈

):

Revision :

6. Instantaneous Acceleration (Acceleration) (𝒂):

direction of 𝒂 :

not related to path

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Phys 110

1. to identify the launched angle of a projectile that is measured from the horizontal.

2. to resolve the initial velocity of the projectile into its components and write it in unit-vector notation.

3. to analyze the projectile motion into two one-dimensional independent motions: horizontal and vertical.

4. to identify the horizontal and vertical components of the acceleration of the projectile.

5. to calculate the horizontal and vertical components of the final velocity of the projectile after time t.

6. to calculate the horizontal and vertical displacement of the projectile after time t.

7. to calculate the maximum height that the projectile can reach.

8. to calculate the time that the projectile spend to reach any position.

9. to define the horizontal Range of the projectile.

10. to calculate the horizontal Range of the projectile.

11. to calculate the maximum horizontal Range of the projectile.

12. to describe the path of the projectile (trajectory).

Chapter 4 : Motion in 2D and 3D

Objectives covered in this lesson :

Motion in 2D and 3D:

Projectile Motion

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Phys 110

4-5 Projectile Motion :

It’s a

2D

motion

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Phys 110

Q: can we consider the following as projectile motions:

1. a tennis ball in flight.

2. a plane in flight.

3. a duck in flight.

The images show

“Types of Projectile

Motions”.

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In studying the projectile motion in this course, we assume that “air” has no effect on

the projectile.

^y

x θ₀

Projectile’s Path

“trajectory”

Launching point

Landing point Max. height

Projectile

R

The range R

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Phys 110

Chapter 4 : Motion in 2D and 3D

4-5 Projectile Motion : Initial Velocity (𝒗

_𝒐

) :

𝑣_𝑜 is the magnitude of 𝑣 _𝑜 .

𝜃_𝑜 is the angle between 𝑣 _𝑜 and the positive x direction.

Scalar components:

Position vector 𝒓 of the motion

&

Velocity vector 𝒗 of the motion change continuously

Acceleration vector 𝒂 of the motion is constant and is always directed

downwards

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Position vector 𝒓 Velocity vector 𝒗 Phys

110

θ₀ y

x v₀

v v

v

v_0x v_0y

v_x v_y

y x x

y y

y

x y

x

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Divide it into two 1D motions:

One in the x-axis (horizontal motion)

One in the y-axis (vertical motion)

and study them separately.

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Phys 110

Chapter 4 : Motion in 2D and 3D

4-6 Projectile Motion Analyzed :

Projectile Motion

Vertical Motion (y-axis) Horizontal Motion (x-axis)

No acceleration

(𝒂

_𝒙

= 𝟎) Free-fall motion

(𝒂

_𝒚

= 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 = −𝒈)

Substitute:

𝒙 → 𝒚

𝒂 → 𝒂_𝒚 → (−𝒈) 𝒗_𝒐 → 𝒗_𝒐𝒚 → 𝒗_𝒐𝒔𝒊𝒏𝜽_𝒐

𝒗 → 𝒗_𝒚 Substitute:

𝒂 → 𝒂_𝒙 → (𝟎) 𝒗_𝒐 → 𝒗_𝒐𝒙 → 𝒗_𝒐𝒄𝒐𝒔𝜽_𝒐

𝒗 → 𝒗_𝒙 𝑣_𝑥 = 𝑣_𝑜𝑥

𝑥 − 𝑥_𝑜 = 𝑣_𝑜𝑥𝑡 𝑣_𝑜𝑥 = 𝑣_𝑜𝑐𝑜𝑠𝜃_𝑜

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Phys 110

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Phys 110

the motion in x the motion in y

substitute 𝒙_𝒐 = 𝟎 , 𝒚_𝒐 = 𝟎 then, combine by eliminating ( t )

Note:

When solving problems: always put the

origin of the xy-graph at the start of the

motion, so that 𝒙

_𝒐

= 𝟎 , 𝒚

_𝒐

= 𝟎.

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Phys 110

the motion in x the motion in y

The Range (R): is the horizontal distance the projectile has travelled when it returns to its initial (launch) height.

substitute 𝒙 − 𝒙_𝒐 = 𝑹 , 𝒚 − 𝒚_𝒐 = 𝟎 then, combine by eliminating ( t ) and using 𝒔𝒊𝒏 𝟐𝜽 = 𝟐 𝒔𝒊𝒏 𝜽 𝒄𝒐𝒔 𝜽

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Phys 110

Chapter 4 : Motion in 2D and 3D

Note: If the projectile’s final height is not the same as its initial height:

The horizontal range ≠ the horizontal distance travelled

The maximum horizontal range ≠ the maximum horizontal distance travelled

R

Horizontal distance y

R x

Horizontal distance

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Phys 110

Chapter 4 : Motion in 2D and 3D

Maximum Height:

x H

Max. height

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Answer: (a) 𝑣_𝑥 is constant.

(b) 𝑣_𝑦 is initially positive, decreases to zero, and then becomes progressively more negative.

(c) 𝑎_𝑥 = 0 throughout the motion.

(d) 𝑎_𝑦 = −𝑔 throughout the motion.

4-6 Projectile Motion Analyzed :

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Problem 21 :

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Problem 21 :

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Phys 110

Problem 38 :

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Phys 110

Problem 38 :

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Phys 110

Chapter 4 : Motion in 2D and 3D

Sample Problem (4-7) :

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Phys 110

Sample Problem (4-7) :

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Motion in 2D and 3D:

Projectile Motion.

Projectile Motion Analyzed.

Phys 110

Lesson 4 of 5 Slide 24 (last)

Summary:

Any Questions?

Next lesson we will cover:

Section (4-7).

Sample problem (4-10).

General Physics 1 (Phys 110 : Mechanics)