Unit 4: Two-Dimensional Kinematics

Section A: Projectile Motion  Corresponding Book Sections:  4.1, 4.2

 PA Assessment Anchors  S11.C.3.1

Difference between 1-D and 2-D  One Dimension  Up / Down  Back / Forth  Left / Right  Example:  Driving a car down a straight street  Two dimension  Projectiles  Vertical & Horizontal motion  Example:  Throwing something up in the air to someone else

Projectile Motion  Motion of objects that are launched  Objects continue moving under only the influence of gravity.

Basic assumptions of this unit… 1. Horizontal and Vertical motions are independent  In other words…treat the horizontal motion as if the vertical motion weren’t there, and vice-versa  You may need to use quantities in both directions, but you treat them separately (i.e.: Separate equations)

Basic assumptions of this unit… 2. Ignore air resistance  We all know that air resistance exists, but to make our lives easier, we’re going to ignore it  Otherwise, the problems get too hard!!

Basic assumptions of this unit… 3. We also ignore the rotation of the Earth  If we were to include the rotation of the Earth, we’d need to include that force in all of the problems…and why would we want to do that? 

Basic assumptions of this unit… 4. The acceleration of gravity is always 9.8 m/s 2 in the downward direction and pulls  This is the same from the last unit. Just remember, if:   You say ↑ is positive, g is negative You say ↑ is negative, g is positive

Basic assumptions of this unit… 5. Gravity only affects the motion in the y-direction and has no effect on the x-direction.

 Think about it…if we’re analyzing the motion separately (vertical and horizontal), when we look at the horizontal motion, gravity doesn’t affect that motion.

The basic kinematics equations… 2-D

Getting Components for the Equations  The equations are the same, they just analyze the x and y directions separately  Remember from vectors:

A

x

= A cos

θ

A

y

= A sin

θ

so......

v ox = v o cos θ v oy = v o sin θ

Two ways to solve the turtle problem...

Method #1 Using vector principles 1 m Problem: How far has the turtle traveled in 5 s (both x and y dir)?

Two ways to solve the turtle problem...

Method #2 Using kinematics equations = .2 m/s Problem: How far has the turtle traveled in 5 s (both x and y dir)?

Practice Problem #1  Refer to Example 4-1 on page 79

Practice Problem #2  Refer to Example 4-2 on Page 80

Section B: Zero Launch Angle  Corresponding Book Sections:  4.3

 PA Assessment Anchors  S11.C.3.1

Zero Launch Angle  Projectile launched horizontally  In other words, the angle between initial velocity and horizontal is 0 °  Projectile has no acceleration in the x direction unless specified  Initial velocity is only in x direction.

Practice Problem #1  A person is walking with a speed of 1.3 m/s and drops a ball he is holding. The ball falls from a height of 1.25 m. Find the horizontal position of the ball after 0.5 s.

Practice Problem #2  A ball is thrown horizontally at 22.2 m/s from the roof of a building. It lands 36 m away from the building. How tall is the building?

Practice Problem #3  A diver running at 1.6 m/s dives out horizontally from the edge of a vertical cliff and reaches the water below 3.0 s later. How high was the cliff and how far from the base did the diver hit the water?

Section C: General Launch Angle  Corresponding Book Sections:  4.4

 PA Assessment Anchors  S11.C.3.1

General Launch Angle  A particle launched at some angle above the horizontal  These are considerably more difficult than the zero-launch angle problem

What is different?

 We need to break the initial velocity into x and y directions.

v ox = v o cos θ v oy = v o sin θ  We may need to use the quadratic equation to solve for time 

Quadratic Equation  Use when solving for time in 2 nd equation:

ax

2 

bx



c

 0 

b



b

2  4

ac

2

a

Practice Problem #1  Refer to Easi-Teach file