Lecture Outlines Chapter 4 Physics, 3

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Lecture Outlines Chapter 4

Physics, 3

Edition

James S. Walker

Chapter 4

Two-Dimensional

Kinematics

Units of Chapter 4

• Motion in Two Dimensions

• Projectile Motion: Basic Equations

• Zero Launch Angle

• General Launch Angle

• Projectile Motion: Key Characteristics

4-1 Motion in Two Dimensions

If velocity is constant, motion is along a

straight line:

Example 4-1

The Eagle Descends

4-1 Motion in Two Dimensions

Motion in the x- and y-directions should be

solved separately:

Example 4-2

A Hummer Accelerates

4-2 Projectile Motion: Basic Equations

Assumptions:

• ignore air resistance

• g = 9.81 m/s ² , downward

• ignore Earth’s rotation

If y-axis points upward, acceleration in

x-direction is zero and acceleration in

y-direction is -9.81 m/s ²

4-2 Projectile Motion: Basic Equations

The acceleration is independent of the direction

of the velocity:

4-2 Projectile Motion: Basic Equations

These, then, are the basic equations of

projectile motion:

Figure 4-3

Independence of vertical and horizontal motions

4-3 Zero Launch Angle

Launch angle: direction of initial velocity with

respect to horizontal

4-3 Zero Launch Angle

In this case, the initial velocity in the y-direction

is zero. Here are the equations of motion, with

x ₀ = 0 and y ₀ = h:

4-3 Zero Launch Angle

This is the trajectory of a projectile

launched horizontally (v ₀ =5 m/s):

Example 4-3

Dropping a Ball

Conceptual Checkpoint 4-1

Compare Splashdown Speeds

4-3 Zero Launch Angle

Eliminating t and solving for y as a function of x:

This has the form y = a + bx ² , which is the equation of a parabola.

The landing point can be found by setting

y = 0 and solving for x:

Example 4-4

Jumping a Crevasse

4-4 General Launch Angle

In general, v _0x = v ₀ cos θ and v _0y = v ₀ sin θ

This gives the equations of motion: (x ₀ =y ₀ =0)

Figure 4-6

Projectile with an arbitrary

launch angle

4-4 General Launch Angle

Snapshots of a trajectory; red dots

are at t = 1 s, t = 2 s, and t = 3 s

Example 4-5

A Rough Shot

Example 4-6

A Leap of Faith

Figure 4-8

Range of a projectile

4-5 Projectile Motion: Key Characteristics

Range: the horizontal distance a projectile travels

If the initial and final elevation are the same:

4-5 Projectile Motion: Key Characteristics

The range is a maximum when θ = 45°:

Figure 4-9

Projectiles with air resistance

Symmetry in projectile motion:

4-5 Projectile Motion: Key Characteristics

Figure 4-10

Velocity vectors for a projectile launched at the origin

Conceptual Checkpoint 4-3

Compare Landing Speeds

Figure 4-11

Range and launch angle in the absence of air resistance

Example 4-7

What a Shot!

Summary of Chapter 4

• Components of motion in the x- and y- directions can be treated independently

• In projectile motion, the acceleration is –g

• If the launch angle is zero, the initial velocity has only an x-component

• The path followed by a projectile is a parabola

• The range is the horizontal distance the

projectile travels