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 and pulls in the downward direction  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 v ox = v o cos θ v oy = v o sin θ so......

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

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

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

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

Practice Problem #1  Refer to Easi-Teach file