Motion in Two Dimensions Use Newton’s laws and your knowledge of vectors to analyze motion in two dimensions. Solve problems dealing with projectile motion. Solve relative velocity problems. Chapter 6 In this chapter you will:
Table of Contents Chapter 6 Chapter 6: Motion in Two Dimensions Section 6.1: Projectile Motion Section 6.3: Relative Velocity
Projectile Motion Recognize that the vertical and horizontal motions of a projectile are independent. Relate the height, time in the air, and initial vertical velocity of a projectile using its vertical motion, and then determine the range using the horizontal motion for all 3 categories of projectile motion Determine maximum height of a projectile in motion Explain how the trajectory of a projectile depends upon the frame of reference from which it is observed. Section 6.1 In this section you will:
Projectile Motion If you observed the movement of a golf ball being hit from a tee, a frog hopping, or a free throw being shot with a basketball, you would notice that all of these objects move through the air along similar paths, as do baseballs, arrows, and bullets. Each path is a curve that moves upward for a distance, and then, after a time, turns and moves downward for some distance. You may be familiar with this curve, called a parabola, from math class. Section 6.1 Projectile Motion
Projectile Motion An object shot through the air is called a projectile. Section 6.1 Projectile Motion • A projectile can be a football, a bullet, or a drop of water. • You can draw a free-body diagram of a launched projectile and identify all the forces that are acting on it. • No matter what the object is, after a projectile has been given an initial thrust, if you ignore air resistance, it moves through the air only under the force of gravity. • The force of gravity is what causes the object to curve downward in a parabolic flight path. Its path through space is called its trajectory.
Independence of motion demonstration, simulations Books Shooting the monkey simulation
Projectile Motion Section 6.1 Independence of Motion in Two Dimensions Click image to view movie.
They should hit at the same time Assuming no air resistance Because… Both have the same vertical acceleration Both start at the same vertical height Both start with the same initial vertical velocity
Key fact –Independence of motion The motion in each dimension is independent of the others Horizontal motion does not effect vertical motion Could have different horizontal accelerations, velocities, displacements and applied net forces, BUT SAME VERTICAL RESULTS
Assumptions That air applies no force to the projectile as it moves through the air A safe assumption at low speeds, or with objects that have significant mass and small surface areas
Acceleration Givens Because of no air and gravitational pull Horizontal acc = 0 (zero) Vertical Acc = (-9.8) (acceleration due to gravity) FOR EVERY PROBLEM
Real life projectiles For fast moving projectiles on Earth, several other things can affect the range and speed of projectiles than gravity: Air resistance (tends to slow down) Spin Forces (can significantly change time in air)
Types of projectile motion There are 3 main categories of projectile motion, based on: Initial set-up Flight path Type 1 (Cliffs) Type 2 (Soccer Pitch) Type 3 (Soccer ball off a cliff)
The Range Means the same as horizontal displacement from launch site to point of impact
Type 1 What information is automatically given for a type 1 problem? Give an example of a type one projectile not connected to running a car off a cliff
Type one answers Both accelerations Vertical initial velocity = 0 A person running off a dock A person shooting the gun horizontally
Recap • Assume only force on projectile is_____ • Shape of trajectory of projectile is ____ • Independence of motion allows us to____ • What information will always be given____ • Why create a chart?
Type 1 (Cliffs) Participant: Initial velocity __________ Height of cliff _________
Questions How much time does it take for the car to reach the ocean? How far from the base of the cliff will the car impact the water? What are the final horizontal and vertical velocities the instant before impact?
Givens Identify the type of value (velocity, displacement, time and/or acceleration) AND Whether it is connected to the horizontal or vertical motion Acceleration is always given: Vertical acceleration (-9.8 m/s2) Horizontal acceleration (0 m/s2) Horizontal acceleration is zero because there is no net force on the object to change the motion
Organize your information! The best way to organize the large amount of information related to projectile problems is by creating a chart Horizontally: Vf = Vi a 0 -9.8 Vi Time is the same in both columns
How to solve for unknowns Since the acceleration for each dimension is constant, you can use the 4 Con A equations to find what you need. The 2 most useful equations will be D = Vit + ½at2 Vf = Vi + at
When Solving… Use either vertical OR horizontal values Do not mix them Time is the only value that is always the same in both columns
When solving Look for the column with 3 known values Generally start with vertical values Warning: Watch out for solving with 3 horizontal values where initial and final velocity are the same Things will cancel out in Vf = Vi + at
Hidden variable(s) for Type 1 projectile problems Besides accelerations The initial vertical velocity is zero
Back to the problem Create chart Identify what you are given Identify what you want Check units Equation Solve Check answer
Last question in class A marble rolls off the edge of a 1.29 m high table with a speed of 4.9 m/s. Complete the chart
Givens H V a 0 -9.8 d X -1.29 vi 4.9 0 vf 4.9 X t X X
Answers H V a 0 -9.8 d 2.51 -1.29 vi 4.9 0 vf 4.9 -4.9 t 0.51 0.51
Type 2 projectile problems Soccer ball problems
Projectile Motion When a projectile is launched at an angle, the initial velocity has a vertical component as well as a horizontal component. If the object is launched upward, like a ball tossed straight up in the air, it rises with slowing speed, reaches the top of its path, and descends with increasing speed. Section 6.1 Projectiles Launched at an Angle
Projectile Motion The adjoining figure defines two quantities associated with a trajectory. One is the maximum height, which is the height of the projectile when the vertical velocity is zero and the projectile has only its horizontal-velocity component. Section 6.1 Projectiles Launched at an Angle
Projectile Motion The other quantity depicted is the range, R, which is the horizontal distance that the projectile travels. Not shown is the flight time, which is how much time the projectile is in the air. For football punts, flight time often is called hang time. Section 6.1 Projectiles Launched at an Angle
What is different between type one and type two Type 1 No initial vertical velocity Final vertical displacement does NOT = zero Type 2 There is initial vertical velocity = component of overall initial velocity Final vertical displacement DOES = zero
Added step in type 2 Break initial velocity up into vertical and horizontal components
Projectile Motion Section 6.1 The Flight of a Ball A ball is launched at 4.5 m/s at 66° above the horizontal. What are the maximum height and flight time of the ball?
Section Projectile Motion 6.1 The Flight of a Ball Establish a coordinate system with the initial position of the ball at the origin.
Section Projectile Motion 6.1 The Flight of a Ball Draw a motion diagram showing v, a, and Fnet. Known: yi = 0.0 m θi = 66° vi = 4.5 m/s ay = −g Unknown: ymax = ? t = ?
Projectile Motion Are the units correct? Dimensional analysis verifies that the units are correct. Do the signs make sense? All should be positive. Are the magnitudes realistic? 0.84 s is fast, but an initial velocity of 4.5 m/s makes this time reasonable. Section 6.1 After calculation, check
Type 3 Projectile problems • Comparisons to type 1 and 2
Example • Joe kicks a 2.3 kg soccer ball off a 25 m high cliff at an initial angle of 48˚ to horizontal • How long does it take the ball to reach the bottom of the cliff? • Range? • Vertical and horizontal components of the final velocity?