ChemPhys 173/273

Unit 7: Vectors and Projectiles

Problem Set C

Overview:

Problem Set C targets your ability to combine a conceptual understanding of projectile motion with an ability to use kinematic equations in order to solve horizontally launched projectile problems. More than ever, you will have to rely upon good problem-solving skills to be successful.

The Known and Unknown Variables

It is suggested that you utilize an "x-y table" to organize your known and unknown information. An x-y table lists kinematic quantities in terms of horizontal and vertical components of motion. The horizontal displacement, initial horizontal velocity. and horizontal acceleration are all listed in the same column. A separate column is used for the vertical components of displacement, initial velocity and acceleration. In this problem set, you will have to give attention to the following kinematic quantities and their corresponding symbols.

 Quantity Symbol Quantity Symbol horizontal displacement x vertical displacement y original horiz. velocity vox original vertical velocity voy horizontal acceleration ax vertical acceleration ay final horizontal velocity vfx final vertical velocity vfy time t

Given these symbols for the basic kinematic quantities, an x-y table for a projectile problem would have the following form:

 Horizontal Vertical x = __________________ vox = __________________ ax = __________________ vfx = __________________ t = __________________ y = __________________ voy = __________________ ay = __________________ vfy = __________________ t = __________________

The Formulas

Of the nine quantities listed above, eight are vectors which have a specific direction asscoiated with them. Time is the only quantitiy which is a scalar. As a scalar, time can be listed in an x-y table in either the horizontal or the vertical columns. In a sense, time is the one quantity which bridges the gap between the two columns. While horizontal and vertical components of motion are independent of each other, both types of quantities are dependent upon time. This is best illustrated when inspecting the kinematic equations which are used in projectile motion problems.

If the understanding that a projectile is an object upon which the only force is gravity is applied to these projectile situations, then it is clear that there is no horizontal acceleration. Gravity only accelerates projectiles vertically, so the horizontal acceleration is 0 m/s/s. Any term containing the ax variable will thus cancel. The three equations in the top row simplify to the following:

Problem Set C focuses on horizontally launched projectiles. A horizontally-launched projectile is an object which initiates its motion by moving only in the horizontal direction. Once such an object becomes a projectile, it maintains the same horizontal motion while accelerating vertically. Since it is initially moving only in the horizontal direction, such a projectile has an original vertical velocity (voy) of 0 m/s. Any term containing the voy variable will thus cancel. For horizontally-launched projectiles, the three vertical or y-equations simplify to the following:

The Basic Strategy

The basic approach to solving horizontally launched projectile problems involves reading the problem carefully and visualizing the physical situation. A well-constructed diagram is an often useful means of visualizing the situation. Then list and organize all known and unknown information in terms of the symbols used in the projectile motion equations. An x-y table is a useful organizing scheme for listing such information. Inspect all known quantities, looking for either three pieces of horizontal information or three pieces of vertical information. Since all kinematic equations list four variables, knowledge of three variables allows you to determine the value of a fourth variable. For instance, if three pieces of vertical information are known, then the vertical equations can be used to determine a fourth (and a fifth) piece of vertical information. Often times, the fourth piece of information is the time. In such instances, the time can then be combined with two pieces of horizontal information to calculate another horizontal variable using the

Habits of an Effective Problem-Solver:

An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach, they all have habits which they share in common. These habits are described briefly here. An effective problem-solver ...

• ... reads the problem carefully and develops a mental picture of the physical situation. If needed, they sketch a simple diagram of the physical situation to help visualize it (e.g., a trajectory diagram).
• ... identifies the known and unknown quantities in an organized manner, often times recording them on the diagram iteself. They equate given values to the symbols used to represent the corresponding quantity (e.g., dx = 24.9 m; dy = -4.5 m, vox ???). In the case of projectiles, it is very useful to use a x-y table.
• ... plot a strategy for solving for the unknown quantity; the strategy will typically center around the use of kinematic equations in a two-dimensional situation.
• ... identify the appropriate formula(s) to use, often times writing them down. Where needed, they perform the needed conversion of quantities into the proper unit.
• ... perform substitutions and algebraic manipulations in order to solve for the unknown quantity.

To be successful on this problem set, you will need to be able to:

• give attention to units, performing proper unit conversions where necessary.
• carefully read and interpret a projectile problem statement, extracting the explicitly stated and implied information relevant to the solution.
• employ the habits of a good problem-solver.

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

What is a Projectile? | Characteristics of a Projectile's Trajectory | Horizontal Velocity Components

View Sample Problem Set.

 Problem Description Audio Link 1 Determination of the time required for a stone to fall vertically from a cliff of known height. 2 Determination of the time required for a stone to fall vertically if thrown from a cliff of known height. 3 Referring to the previous problem; determination of the horizontal displacement of the stone. 4 Determination of the horizontal displacement of a cat who runs off a table with a known speed from a known height. 5 Referring to the previous problem; determination of the horizontal component of the final velocity. 6 Referring to the previous problem;determination of the vertical component of the final velocity. 7 Determination of the height of diving platform from the time required for a watermelon to fall vertically to the water below. 8 Referring to the previous problem; determination of the horizontal displacement of the watermelon. 9 Referring to the previous problem; determination of the final speed of the watermelon. 10 Determination of the horizontal speed of some cliff divers from the height of the cliff and the horizontal displacement. 11 Analysis of two balls - one dropped from rest and the other launched horizontally from the same height; must determine the original separation distance from the launch speed and the time of fall. 12 Determination of the initial speed of a ball which rolls off a table of known height and lands a known horizontal distance from its base. 13 Referring to the previous problem; determination of the acceleration of the ball at the halfway point to the ground. 14 Referring to the previous problem; determination of the final speed of the ball. 15 Determination of the height of a table from knowledge of the speed at which a Hot Wheels car rolls off it and the horizontal distance which it lands from its edge. 16 Determination of the horizontal displacement of a dropped care package from knowledge of the plane height and the horizontal air speed. 17 Determination of the initial horizontal speed of a thrown ball if knowledge of its vertical and horizontal displacements are known. 18 An angle-launched projectile problem; determination of the time of flight if given the initial launch velocity and angle. 19 Referring to the previous problem; determination of the horizontal displacement of the projectile. 20 Referring to the previous problem; determination of the peak height of the projectile.

Audio Help for Problem: 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20

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