Lecture Outlines

Chapter 4

Physics, 3rd Edition

J S W lkJames S. Walker

© 2007 Pearson Prentice Hall

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Chapter 4Chapter 4

Two DimensionalTwo-Dimensional KinematicsKinematics

Units of Chapter 4

• Motion in Two Dimensions

• Projectile Motion: Basic Equations

• Zero Launch Angle

• General Launch AngleGeneral Launch Angle

• Projectile Motion: Key Characteristics

4-1 Motion in Two Dimensions

If velocity is constant, motion is along a straight line:straight line:

4-1 Motion in Two Dimensions

Motion in the x- and y-directions should be solved separately:solved separately:

4-2 Projectile Motion: Basic Equations

Assumptions:

• ignore air resistance

• g = 9 81 m/s2 downward• g = 9.81 m/s2, 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/s2

4-2 Projectile Motion: Basic Equations

The acceleration is independent of the direction of the velocity:of the velocity:

4-2 Projectile Motion: Basic Equations

These, then, are the basic equations of projectile motion:projectile motion:

4-3 Zero Launch AngleLaunch 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, withis zero. Here are the equations of motion, with x0 = 0 and y0 = h:

4-3 Zero Launch AngleThis is the trajectory of a projectile launched horizontally:

4-3 Zero Launch AngleEliminating t and solving for y as a function of x:

This has the form y = a + bx2, which is the equation of a parabola.

The landing point can be found by setting y = 0 and solving for x:

4-4 General Launch Angle

In general, v0x = v0 cos θ and v0y = v0 sin θ

This gives the equations of motion:This gives the equations of motion:

4-4 General Launch AngleSnapshots of a trajectory; red dots are at t = 1 s, t = 2 s, and t = 3 s

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°:

4-5 Projectile Motion: Key Characteristics

Symmetry in projectile motion:

Summary of Chapter 4

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

• In projectile motion, the acceleration is –g

If h l h l i h i i i l l i• 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