Vector Decomposition

Transcription

1 Projectile Motion AP Physics 1 Vector Decomposition 1

2 Coordinate Systems A coordinate system is an artificially imposed grid that you place on a problem. You are free to choose: Where to place the origin, and How to orient the axes (you can rotate the axis if you wish, which is sometimes beneficial). To the right is a conventional xy-coordinate system and the four quadrants I through IV. We will discuss the most appropriate choices of coordinate system for each problem that we encounter in this class. Vector Components In many units we will need to break a vector into components. This is the reverse process of vector addition (this is often called vector decomposition). You can break a vector into as many pieces as you want, be we generally just want one component that is parallel to our x-axis (we call this our x component) and one component that is parallel to the y axis (we call this our y component). 2

3 Vector Decomposition To accomplish vector decomposition, we will use basic right triangle trig. This will be a very important tool in all of physics, particularly with projectiles and forces. Example Determine the x and y components of the vectors below. 3

4 Example Find the x- and y-components of the acceleration vector a shown below. Basics of Projectiles 4

5 Projectile Motion Projectile motion is an extension to two dimensions of free-fall motion. A projectile is an object that moves in two dimensions under the influence of gravity and nothing else. As long as we can neglect air resistance, any projectile will follow the same type of path. The path of a projectile looks like a parabola! A Simple Experiment Consider the following experiment: We take two identical bullets and load one of them into a gun. Starting both bullets from the same height, we simultaneously fire one bullet and drop the other. If the fired bullet was shot horizontally, which bullet will strike the ground first? 5

6 What the What?!? THEY HIT AT THE SAME TIME! This fact teaches us a very important concept behind projectiles: The vertical and horizontal motion of a projectile are INDEPENDENT of each other. The projectile does not know if it is being launched or dropped, nor does it care. The horizontal and vertical motion do not affect one another. Because of this, we will analyze projectiles in the horizontal and vertical directions separately. This means that we will be breaking vectors (mostly velocity) into x and y components. Horizontal Velocity Component The projectile covers equal horizontal displacements in equal time periods. This means the initial horizontal velocity equals the final horizontal velocity. In other words, the horizontal velocity is constant. BUT WHY? The projectile only accelerates because of gravity, which points down. Because of this, the horizontal component of velocity is unaffected. The acceleration of a projectile in the horizontal direction is 0! 6

7 Vertical Velocity Component There is acceleration in the vertical direction, which means that a projectile does not cover equal displacements in equal periods of time. For a projectile launched at an angle, both the magnitude and direction of the vertical velocity change. On the way up, the vertical velocity decreases in magnitude. At the very top of the flight, the vertical velocity is equal to 0 (this is a very important aspect of projectiles that can make the math much easier). On the way down, the vertical velocity points down and increases in magnitude. Return of the Kinematics Because projectile problems focus mostly on position, velocity, acceleration and time, we will use our kinematic equations to analyze projectile motion. The process is much the same as with 1-D kinematics problems, only now we may have to analyze the horizontal and vertical direction in a single problem. 7

8 Types of Launches Horizontal Launch This is the simplest kind of projectile that you can have. For a horizontally launched projectile, we will always know a few things: The initial y velocity is equal to 0. The x velocity is constant. 8

9 Time of Flight Horizontal Launch A common projectile question is to ask for the time of flight for a horizontally launched projectile. Given that the initial y velocity is equal to 0 for such a launch, we can simplify the second kinematic to solve for time. t = 2H g This is ONLY FOR HORIZONTAL LAUNCH. Don t use it for a launch at an angle. H represents launch height. g is 9.8 m/s 2, don t add a negative to this equation. If you do that you end up with an imaginary number for your time. Lets not do that. Example A plane traveling with a horizontal velocity of 100 m/s is 500 m above the ground. At some point the pilot decides to drop some supplies to designated target below. a. How long is the drop in the air? b. How far horizontally does the drop travel from the moment it is released to the moment it hits the ground? c. How far horizontally does the plane travel from the moment the drop is released to the moment it hits the ground? 9

10 Projectiles at Angles Projectiles can also be launched at angles, which requires a little more care than if launched horizontally. Unlike with a horizontal projectile, we DO have an initial vertical velocity when launched at an angle. Because of this we will generally have to start by breaking up the initial velocity into vertical and horizontal components. Angled projectiles can be launched either ground to ground (no net elevation change), or elevated launch. Breaking Up Initial Velocity Breaking up the initial velocity into horizontal and vertical components allows us to make our list of known and unknown values. All we need to break up the velocity is the launch speed and the angle of launch. Use trigonometry to find the horizontal and vertical components. 10

11 Ground to Ground (GTG) Launch If a projectile is launched from a certain height and comes back down to that same height, we can use the symmetry of flight to analyze the motion. Because the y velocity is 0 at the top of the flight (often called the apex), we can break the problem into on the way up, and on the way down. When working with a GTG projectile, it is often easiest to define the ground to be a height of 0. Example A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees. a. How long is the ball in the air? b. How far away does it land? c. How high does it travel? 11

12 Example Projectiles 1 and 2 are launched over level ground with the same speed but at different angles. Which hits the ground first? Ignore air resistance. Projectiles 1 and 2 are launched over level ground with different speeds. Both reach the same height. Which hits the ground first? Ignore air resistance. Range of a GTG Launch The range of a projectile is the horizontal distance traveled. Without air resistance, the maximum range for a projectile is achieved at a 45 launch angle. This is only true if the launch is GTG. For uneven launch (we will see next), the angle is less than

13 Elevated Launch Angled projectiles can also have a launch height that is different from the landing height. For these kinds of launch it is often easiest to define the lowest point on the path to be a height of 0. This prevents us from having a negative position. These kinds of problems do not show up often, so there will not be a heavy emphasis on them. Example A projectile is launched from the top of a cliff. The cliff is 100 m high, and the projectile is launched from the cliff in the direction of the level plane below. At launch, the projectile has a velocity of 70 m/s at an angle 25 o above the horizontal. Air resistance is negligible. a. Calculate the total time from launch until the projectile hits the ground. b. Calculate the horizontal distance that the projectile travels before it hits the ground c. Calculate the speed at impact. 13

14 One More Experiment A hunter is walking through the jungle when he sees a monkey hanging from a tree branch. The hunter decides he wishes to subdue the monkey to take him home and teach him the fundamentals of quantum field theory. However, right as the hunter pulls the trigger (it s a tranquilizer, no monkeys were harmed in the making of this hypothetical scenario), the monkey lets go of the tree branch. In order to hit the monkey, does the hunter need to aim right at the monkey, below the monkey, or above the monkey? 14