Ch3_souflige

toc Chapter 3 Projectile Motion

=Labs & Activities=

Vector Addition Activity
10.19

Note: Experimental Resultant = 19.08 m

Graphical Method of Vector Addition R=20.72m@35 degrees SW

Analytical Method of Vector Addition R=20.55m@33.5 degrees SW



Percent Error: 7.2% error for analytical method 7.9% error for graphical method

Ball in Cup
10.24.11

Calculations. //Range = 3.16 m// //Hang Time = 0.86 m// //Time = 0.42 s// //Velocity Initial = 7.52 m/s//

//Ball in Cup:// //**Predicted** Range = 3.28 m// //Hang time = 0.93 m// //Time = 0.436 s//

Experimental Result: 2.92m

__Percent Error:__



media type="file" key="Movie on 2011-10-25 at 09.15.mov" width="300" height="300"

Conclusion:

Our results were not consistent with our hypotheses. This probably means that we originally incorrectly calculated the velocity as being too fast. With almost an 11% error, the x-distances were 2.92 m (experimental) and 3.28 (theoretical). There were many reasons for this error, the most prominent being rounding issues. Because of this lack of precision, the theoretical range was greatly affected. Also, the lateral inaccuracy posed a problem. Each shot fired did not consistently hit the same lateral position. Unfortunately, this gave varied results. The shooter also shifted slightly with each trial, giving us inconclusive results.

Shoot Your Grade
10.26

Purpose and Rationale: Launch a ball from the launcher at a given angle and speed setting so that the ball passes consecutively through five rings and lands in a cup on the floor.

Materials and Methods: By using the techniques we previously learned we are able to find the initial velocity and time, as well as the x and y displacements (which were also found through measurement). Afterwords, the path of the ball is broken up into 6 equal sections. At each interval the y position is found to judge how high/low the rings should be hung. The materials used for these processes are a launcher, ball, cup, tape measure, five rings, and string with which to hang them.

Observations and Data from Initial Velocity:

The information above fully describes the x and y components of the ball's predicted trajectory. Initial velocity helps to find the projected distances and heights. Acceleration produces the parabolic trajectory. Displacement helps set up intervals.
 * =  ||= X ||= Y ||
 * = Initial Velocity ||= 6.77cos(15) ||= 6.77sin(15) ||
 * = Acceleration ||= 0 ||= -9.8 (m/s/s) ||
 * Time || 0.71 (s) || 0.71 (s) ||
 * Displacement || 4.66 (m) || -1.22 (m) ||

Observations and Data from Performance media type="file" key="Movie on 2011-11-09 at 09.26.mov"

Physics Calculations

Angle measure: 15 degrees Velocity Initial: 6.77 m/s Range: 4.66 m (average) Height: 1.22 m Time: 0.71 s

__Intervals__ 6 sections: 4.66/6 = 0.78 each

Intervals 1 and 5

Interval 2

Interval 3

Interval 4

Interval 6 (cup)

Graph of the ball's projected curve.

The equations used to yield this graph.

Error Analysis


 * Experimental heights:**
 * Interval || Height from ground ||
 * 1 || 1.28 m ||
 * 2 || 1.32 m ||
 * 3 || 1.19 m ||
 * 4 || 1.05 m ||
 * Theoretical heights:**
 * Interval || Height from ground ||
 * 1 || 1.359 m ||
 * 2 || 1.358 m ||
 * 3 || 1.21 m ||
 * 4 || 0.93 m ||

Percent Error Calculations

Conclusion: Was the purpose satisfied?

Unfortunately time constraints did not allow for the testing of all the intervals to fully test our hypothesis. The evidence suggests that our calculations were relatively accurate. However, as the ball traveled further from the launcher, its path became much more unpredictable. This is probably because more outside influences affect its trajectory than just gravity.

Conclusion: Experimental Errors

Percent error was within 6% until interval 4. Many factors led to error. Our crude methods of measuring, marking, and placing rings left a lot of room for error. What's more, the lateral aspect of the ball's path had to be accounted for, which offset some positions. The ball also shot very inconsistently. Therefore, the measurements recorded may have worked once at the time, but that does not necessarily mean they will work every time.

Conclusion: Implications for further discussion Placing the rings on sturdy, adjustable structures held in place on the floor could have made the project much more efficient and useful because of increased accuracy. If we could also add a longer barrel to the launcher, the ball would not undergo as much lateral motion. Lastly, if these said ground-based structures were used for the intervals, the measurements would be much more exact and more comparable to the theoretical data. This idea of projectile motion is often used, for better or worse, in the military when dropping bombs from planes, firing howitzers, etc..

Gord-O-Rama
Top View Side View

Mass: 2.19 kg Time: 2.30 seconds Distance: 6.19 m

Calculations:

∆d=1/2(vf+vo)t 6.19=1/2(0+vo)(2.30) Velocity initial = 5.38 m/s

∆d=vt+1/2at 2 6.19=(5.38)(2.30)+1/2(a)(2.30)^2 a= -2.34 m/s/s

At bottom of ramp:

vf^2=vo^2+2a∆d vf^2=(5.38)^2+2(2.34)(2) 6.19 m/s = velocity final (at bottom of ramp)

Overview and Summary:

My vehicle was too bulky and had poorly-chosen wheels. The wood and nails with which I constructed the body were unnecessarily heavy - better would've been thinner, bendable wood. Also, a major drawback of the vehicle was the wheels. Taken from a wheelie chair, they tended to swivel. Though I remedied this using tape they did not work perfectly. One broke after the first trial. In retrospect, skateboard trucks, bearings, and wheels would've provided a more stable set of wheels.

=Notes=

10.13
Basic Vector Notes

Measuring Angles

Drawing Angles

10.14
Notes on Vector Addition

Notes on Vector addition (cont.)

Graph estimation

10.19
Rowboat Problems

10.21
Horizontally Launched Objects. Notes on range (x), hang time (y) and equations.

Two examples of Horizontally Launched Projectile Problems

10.25
Ground-to-Ground Projectile Motion

Vectors Lesson 1
Quantities used to describe physical world = [|vectors and scalars]. A vector quantity is described by magnitude and direction. A scalar quantity is described by magnitude. Vector quantities are often represented by scaled vector diagrams (arrow drawn to scale in specific direction) The vector diagram depicts a displacement vector, must have scale; direction, magnitude in proportion to scale. Directions of Vectors: Magnitude of Vectors: The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow, directly proportional to scale.
 * A. Vectors and Direction **
 * 1) The direction of a vector is often expressed as an angle of rotation of the vector about its tail from EWNS (ex. 45 degrees NW = 45 degrees north //of// west)
 * 2) The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its tail from due East.

Two vectors can be added together to determine the result (or resultant). //Net force// of object = vector sum of individual forces acting on object. In this unit, summing vectors will be where they are directed in directions not vertical/horizontal. There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors: When the two vectors that are to be added do not make right angles to one another, or when there are more than two vectors to add together, the head-to-tail vector addition method is used. A step-by-step method for applying the head-to-tail method to determine the sum of two or more vectors (resultant) is given below. The resultant will still have the same magnitude and direction regardless of order of addition.
 * B. Vector Addition **
 * the Pythagorean theorem and trigonometric methods
 * the Pythagorean theorem used for determining sum of two (and only two) vectors __that make a right angle__ to each other. (a2+b2=c2). Finds missing side length to right triangle.
 * The direction of a //resultant// vector can be found by use of trigonometric functions (SOH CAH TOA). These three functions (sine, cosine, tangent) relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. Once the measure of the angle is determined, the direction of the vector can be found.
 * the head-to-tail method using a scaled vector diagram
 * 1) Choose a scale.
 * 2) Pick a starting location. Label the magnitude and direction of the scale on the diagram.
 * 3) Starting from where the head of the first vector ends, draw the second vector //to scale// in the indicated direction.
 * 4) Repeat steps 2 and 3 for all vectors that are to be added
 * 5) Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as **Resultant** or simply **R**.
 * 6) Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale.
 * 7) Measure the direction of the resultant using the counterclockwise convention.

10.13.11 -- Vectors Lesson 1c-d


 * C. Resultants **

The ** resultant ** is the vector sum, or //result//, of two or more vectors. Vectors A+B+C = R where R is resultant. When R is resultant displacement, the displacement A+B+C is same as R. Same applies for velocity, force, momentum, etc.. Use vector addition methods to find resultant.


 * D. Vector Components **

Vectors in Unit 3 and hereafter are 2D, no longer 1D. When vectors are not directed at angles to the customary coordinate axes, break into two components. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equivalent to the influence of the single two-dimensional vector.

Vectors at an angle to a horizontal have two (x and y) components. This describes 2D motion. The process of determining the magnitude of a vector is known as ** vector resolution **.
 * E. Vector Resolution **
 * 1) the parallelogram method involves using a vector diagram to determine the components of the vector:
 * 2) Sketch vector.
 * 3) Sketch a parallelogram around the vector: beginning at [|tail] of vector, sketch vertical and horizontal lines; then same at [|head] of vector.
 * 4) Draw the components of the vector. The components are the //sides// of the parallelogram. The tail of the components start at the tail of the vector and stretches along the axes to the nearest corner of the parallelogram. Use arrowheads to indicate direction.
 * 5) Label the components of the vectors with symbols to indicate which component represents which side.
 * 6) Measure the length of the sides of the parallelogram and label magnitude
 * 7)  [|the trigonometric method] involves using trigonometric functions to determine the components of the vector:


 * 1) Sketch vector with labeled magnitude and angle to horizontal.
 * 2) Draw a rectangle about the vector such that the vector is the diagonal of the rectangle.
 * 3) Draw the components of the vector. The components are the //sides// of the rectangle. The tail of each component begins at the tail of the vector and stretches along the axes to the nearest corner of the rectangle. Use arrowheads to indicate direction.
 * 4) Meaningfully label the components of the vectors with symbols to indicate which component represents which side.
 * 5) To determine the length of the side opposite the indicated angle, use the sine function.
 * 6) Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.


 * F. Component Method of Vector Addition **

Vectors forming right angles can be added using Pythagorean Theorem. A right triangle has two sides plus a hypotenuse; so the Pythagorean theorem is perfect for adding two right angle vectors. The addition of more than two vectors doesn’t form right triangle, and oftentimes two vectors do not form right triangles.

__ Addition of 3+ //Right Angle// vectors __ : Vectors being added can be rearranged in such a way that they will form a right triangle. Pythagorean Theorem can then be used.

__ SOH CAH TOA and the Direction of Vectors: __ Use SOH CAH TOA to determine the direction of a resultant. Make sure to use angle that forms with the resultant and one of the vectors. (must form right triangle)

__ Addition of Non-Perpendicular Vectors: __ Use vector components (x and y direction) to replace a non-perpendicular vector since A = Ax + Ay. (use sine and cosine to determine the components). The Pythagorean theorem can then be used to determine the magnitude of the resultant, since it will now form right triangle.

Vectors Lesson 2

 * A. What is a Projectile? **

Preview Questions:

What is a Projectile? An object upon which the only force acting is gravity.

What is projectile motion? Once projected dropped, the motion is continued by the object's own inertia and influenced only by gravity.

How is projectile motion affected by inertia? The motion continues by relying solely on the object's inertia.

Is projectile motion always 2D? No. When it is 2D however, it forms parabolic trajectory, caused by gravity.

Do projectile objects always need to be projected? No. Objects that are dropped or thrown vertically are considered projectile.

Main Idea:

A projectile is an object upon which the only force is gravity, and where its motion is continued only the object's inertia. In 2D motion, projectile form parabolic trajectory.


 * B. Characteristics of a Projectile's Trajectory **

Preview Questions:

What does a projectile's trajectory look like? A parabola.

How can it be found? Using the x-component and y-component. An x vs. y graph would show the parabolic trajectory.

What differences arise between horizontally and non-horizontally launched projectiles? They are the same concept. The object moves horizontally as it also moves upward/downward.

How does gravity affect the path? Gravity pulls the object downwards at 9.8 m/s/s.

What does a path without gravity look like? For horizontal motion: A horizontal line with constant speed. For vertical motion: A free-fall path For an angled motion: Constant slope.

Main Idea: A projectile's trajectory is parabolic because of the influence of gravity: it moves horizontally (x-component) as it moves upward/downward (y-component).

1. Horizontal and Vertical Velocity 2. Horizontal and Vertical Displacement
 * C. Describing Projectiles With Numbers **

Preview Questions:

How is parabolic trajectory measured?

Using the x and y components. The trajectory is symmetrical by nature.

How does the angle of projection affect the graph? It doesn’t. The projectile will always slow down toward its peak, and then speed up on its descent.

How are horizontal and vertical positions measured?

The vertical component has an acceleration of 9.8 m/s/s due to gravity. The horizontal component is measured as constant speed.

What are the equations for finding the x and y components?

x = vix • t

y = viy • t + 0.5 • g • t2

How is the displacement of a projectile found?

Using the x and y components.

Main Idea: Using parametric equations for x and y, the path of a projectile can be graphed as a function of time. The x components and y components can be defined in equations.