In the previous part of Lesson 2the use of kinematic equations to solve projectile problems was introduced and demonstrated. These equations were used to solve problems involving the launching of projectiles in a horizontal direction from an elevated position. In this section of Lesson 2, the use of kinematic equations to solve non-horizontally launched projectiles will be demonstrated. A non-horizontally launched projectile is a projectile that begins its motion with an initial velocity that is both horizontal and vertical.
To treat such problems, the same principles that were discussed earlier in Lesson 2 will have to be combined with the kinematic equations for projectile motion. You may recall from earlier that there are two sets of kinematic equations - a set of equations for the horizontal components of motion and a similar set for the vertical components of motion. For the horizontal components of motion, the equations are. Of these three equations, the top equation is the most commonly used.
The other two equations are seldom if ever used. For the vertical components of motion, the three equations are. In each of the above equations, the vertical acceleration of a projectile is known to be As discussed earlier in Lesson 2the v ix and v iy values in each of the above sets of kinematic equations can be determined by the use of trigonometric functions.
The topic of components of the velocity vector was discussed earlier in Lesson 2. To illustrate the usefulness of the above equations in making predictions about the motion of a projectile, consider their use in the solution of the following problem. Determine the time of flight, the horizontal displacement, and the peak height of the football. The solution of any non-horizontally launched projectile problem in which v i and Theta are given should begin by first resolving the initial velocity into horizontal and vertical components using the trigonometric functions discussed above.
In this case, it happens that the v ix and the v iy values are the same as will always be the case when the angle is degrees. The solution continues by declaring the values of the known information in terms of the symbols of the kinematic equations - x, y, v ixv iya xa yand t. In this case, the following information is either explicitly given or implied in the problem statement :. As indicated in the table, the final x-velocity v fx is the same as the initial x-velocity v ix.
This is due to the fact that the horizontal velocity of a projectile is constant ; there is no horizontal acceleration. The table also indicates that the final y-velocity v fy has the same magnitude and the opposite direction as the initial y-velocity v iy. This is due to the symmetrical nature of a projectile's trajectory. The unknown quantities are the horizontal displacement, the time of flight, and the height of the football at its peak.
The solution of the problem now requires the selection of an appropriate strategy for using the kinematic equations and the known information to solve for the unknown quantities. There are a variety of possible strategies for solving the problem. An organized listing of known quantities in two columns of a table provides clues for the selection of a useful strategy.In the Vector Addition Lab, Anna starts at the classroom door and walks:.
Using either a scaled diagram or a calculator, determine the magnitude and direction of Anna's resulting displacement. To insure the most accurate solution, this problem is best solved using a calculator and trigonometric principles.
The first step is to determine the sum of all the horizontal east-west displacements and the sum of all the vertical north-south displacements. Vertical: The series of five displacements is equivalent to two displacements of 30 meters, West and 4 meters, North. The resultant of these two displacements can be found using the Pythagorean theorem for the magnitude and the tangent function for the direction.
A non-scaled sketch is useful for visualizing the situation. Applying the Pythagorean theorem leads to the magnitude of the resultant R. The angle theta in the diagram above can be found using the tangent function. This angle theta is the angle between west and the resultant. Directions of vectors are expressed as the counterclockwise angle of rotation relative to east.
So the direction is 7. In a grocery store, a shopper walks She then turns left and walks Finally, she turns right and walks 8. This problem is best approached using a diagram of the physical situation. The three displacements are shown in the diagram below on the left. Since the three displacements could be done in any order without effecting the resulting displacement, these three legs of the trip are conveniently rearranged in the diagram below on the right.
Now it is obvious from the diagram on the right that the three displacement vectors are equivalent to two perpendicular displacement vectors of These two vectors can be added together and the resultant can be drawn from the starting location to the final location. A sketch is shown below. Since these displacement vectors are at right angles to each other, the magnitude of the resultant can be determined using the Pythagorean theorem. The work is shown below. This is the angle which the resultant makes with the original line of motion the A hiker hikes The hiker then makes a turn towards the southeast and finishes at the final destination.
The overall displacement of the two-legged trip is Determine the magnitude and direction of the second leg of the trip. Like the previous problem and most other problems in physicsthis problem is best approached using a diagram.
The first displacement is due South and the resulting displacement at degrees is somewhere in the fourth quadrant.One of the powers of physics is its ability to use physics principles to make predictions about the final outcome of a moving object. Such predictions are made through the application of physical principles and mathematical formulas to a given set of initial conditions. In the case of projectiles, a student of physics can use information about the initial velocity and position of a projectile to predict such things as how much time the projectile is in the air and how far the projectile will go.
The physical principles that must be applied are those discussed previously in Lesson 2. The mathematical formulas that are used are commonly referred to as kinematic equations. Combining the two allows one to make predictions concerning the motion of a projectile.
In a typical physics class, the predictive ability of the principles and formulas are most often demonstrated in word story problems known as projectile problems. There are two basic types of projectile problems that we will discuss in this course. While the general principles are the same for each type of problem, the approach will vary due to the fact the problems differ in terms of their initial conditions.
The two types of problems are:. A projectile is launched with an initial horizontal velocity from an elevated position and follows a parabolic path to the ground. Predictable unknowns include the initial speed of the projectile, the initial height of the projectile, the time of flight, and the horizontal distance of the projectile.
Problem Type A projectile is launched at an angle to the horizontal and rises upwards to a peak while moving horizontally. Upon reaching the peak, the projectile falls with a motion that is symmetrical to its path upwards to the peak. Predictable unknowns include the time of flight, the horizontal range, and the height of the projectile when it is at its peak.
The second problem type will be the subject of the next part of Lesson 2. In this part of Lesson 2, we will focus on the first type of problem - sometimes referred to as horizontally launched projectile problems. Three common kinematic equations that will be used for both type of problems include the following:. The above equations work well for motion in one-dimension, but a projectile is usually moving in two dimensions - both horizontally and vertically. Since these two components of motion are independent of each other, two distinctly separate sets of equations are needed - one for the projectile's horizontal motion and one for its vertical motion.
Thus, the three equations above are transformed into two sets of three equations. For the horizontal components of motion, the equations are. Of these three equations, the top equation is the most commonly used. Once this cancellation of ax terms is performed, the only equation of usefulness is:. In each of the above equations, the vertical acceleration of a projectile is known to be Thus, any term with v iy in it will cancel out of the equation. The two sets of three equations above are the kinematic equations that will be used to solve projectile motion problems.
To illustrate the usefulness of the above equations in making predictions about the motion of a projectile, consider the solution to the following problem. The solution of this problem begins by equating the known or given values with the symbols of the kinematic equations - x, y, v ixv iya xa yand t. Because horizontal and vertical information is used separately, it is a wise idea to organized the given information in two columns - one column for horizontal information and one column for vertical information.
In this case, the following information is either given or implied in the problem statement:. As indicated in the table, the unknown quantity is the horizontal displacement and the time of flight of the pool ball. The solution of the problem now requires the selection of an appropriate strategy for using the kinematic equations and the known information to solve for the unknown quantities.
It will almost always be the case that such a strategy demands that one of the vertical equations be used to determine the time of flight of the projectile and then one of the horizontal equations be used to find the other unknown quantities or vice versa - first use the horizontal and then the vertical equation. An organized listing of known quantities as in the table above provides cues for the selection of the strategy. For example, the table above reveals that there are three quantities known about the vertical motion of the pool ball.To create this article, volunteer authors worked to edit and improve it over time.
This article has been viewed 34, times. Learn more Projectile motion is often one of the most difficult topics to understand in physics classes. Most of the time, there is not a direct way to get the answer; you need to solve for a few other variables to get the answer you are looking for.
This means in order to find the distance an object traveled, you might first have to find the time it took or the initial velocity first. Just follow these steps and you should be able to fly through projectile motion problems! Please help us continue to provide you with our trusted how-to guides and videos for free by whitelisting wikiHow on your ad blocker. Log in Facebook. No account yet?
Label the distances and velocities given in the problem on your picture. You should be able to look at the picture and have a clear understanding of the path and values given in the problem. List all your variables. It may seem unnecessary if you have your picture properly labeled, but keeping all your variables together in a list will help show what you are missing.
These variables should include your final velocity, initial velocity, distance, acceleration, and time. Since this is projectile motion problem, however, there are different values for the object in the x and y direction. This means you will need to make two lists. It is important to read the question carefully and label your values accordingly.
Make sure the units match.
What is a Projectile?
Convert your units into meters and seconds so that you can use constants to solve the problem. Use physics constants to fill in some unknown variables. In the y direction, acceleration will always be If your problem is set up like scenario 2make sure to use the angle given to break up the initial velocity into x and y components. If your problem is set up like scenario 1the initial velocity is simply what is given.A set of instructional pages written in an easy-to-understand language and complemented by graphics and Check Your Understanding sections.
An ideal starting location for those grasping for understanding or searching for answers. Designed with tablets such as the iPad and with Chromebooks in mind, this user-friendly section is filled with skill-building exercises, physics simulations, and game-like challenges.
Have you tried a Concept Builder lately? You should. Filled with interactive elements, this section is the perfect tool for getting students thinking about the meaning of concepts. And for Chemistry types, we've even started growing a few Chemistry Concept Builders.
If you don't mind a little pixel dust, visit our newest section. Started in August ofour Video Tutorial provides a video-based alternative to the written Tutorial above.
So put on your hard hat and enjoy the shows A large collection of GIF animations and QuickTime movies designed to demonstrate physics principles in a visual manner. Each animation is accompanied by explanations and links to further information. A collection of pages which feature interactive Shockwave files that simulate a physical situation.
Users can manipulate a variable and observe the outcome of the change on the physical situation. We have completed Version 2. Teachers will appreciate the extensive progress reports provided by the App version of our Minds On Physics program.
Students will enjoy using these for practice and teachers can use them as homework assignments. With problems, answers and solutions, The Calculator Pad offers the beginning student of physics the opportunity to conquer the most dreaded part of a physics course - physics word problems.
Each problem is accompanied by a concealed answer which can be revealed by clicking a button. And each audio-guided solution not only explains how to solve the particular problem, but describes habits which can be adopted for solving any problem.
Each review complements a chapter from The Physics Classroom Tutorial. A variety of question-and-answer pages which target specific concepts and skills. Topics range from the graphical analysis of motion and drawing free body diagrams to a discussion of vectors and vector addition. Calling all high school juniors: You've trusted The Physics Classroom to help prepare you for that unit exam in physics.In Unit 1 of the Physics Classroom Tutorial, we learned a variety of means to describe the 1-dimensional motion of objects.
In Unit 2 of the Physics Classroom Tutorial, we learned how Newton's laws help to explain the motion and specifically, the changes in the state of motion of objects that are either at rest or moving in 1-dimension. Now in this unit we will apply both kinematic principles and Newton's laws of motion to understand and explain the motion of objects moving in two dimensions.
The most common example of an object that is moving in two dimensions is a projectile. Thus, Lesson 2 of this unit is devoted to understanding the motion of projectiles. A projectile is an object upon which the only force acting is gravity. There are a variety of examples of projectiles. An object dropped from rest is a projectile provided that the influence of air resistance is negligible.
An object that is thrown vertically upward is also a projectile provided that the influence of air resistance is negligible. And an object which is thrown upward at an angle to the horizontal is also a projectile provided that the influence of air resistance is negligible.
A projectile is any object that once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity. By definition, a projectile has a single force that acts upon it - the force of gravity.
If there were any other force acting upon an object, then that object would not be a projectile. Thus, the free-body diagram of a projectile would show a single force acting downwards and labeled force of gravity or simply F grav. Regardless of whether a projectile is moving downwards, upwards, upwards and rightwards, or downwards and leftwards, the free-body diagram of the projectile is still as depicted in the diagram at the right. By definition, a projectile is any object upon which the only force is gravity.
Many students have difficulty with the concept that the only force acting upon an upward moving projectile is gravity. Their conception of motion prompts them to think that if an object is moving upward, then there must be an upward force. And if an object is moving upward and rightward, there must be both an upward and rightward force.
Their belief is that forces cause motion; and if there is an upward motion then there must be an upward force. They reason, "How in the world can an object be moving upward if the only force acting upon it is gravity? Newton's laws suggest that forces are only required to cause an acceleration not a motion.
Recall from the Unit 2 that Newton's laws stood in direct opposition to the common misconception that a force is required to keep an object in motion. This idea is simply not true! A force is not required to keep an object in motion.
A force is only required to maintain an acceleration. And in the case of a projectile that is moving upward, there is a downward force and a downward acceleration. That is, the object is moving upward and slowing down. To further ponder this concept of the downward force and a downward acceleration for a projectile, consider a cannonball shot horizontally from a very high cliff at a high speed.I did find last week's winning lottery numbers on page 18 though.
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Projectile Problems with Solutions and Explanations
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