Physics 163

Unit 3: Motion in Two-Dimensions

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. Such skills include:

• reading the problem carefully and diagramming the physical situation
• identifying known and unknown information in an organized manner
• taking the time to plot out a strategy prior to beginning the solution
• identifying an appropriate formula to use
• performing step-by-step algebraic manipulations

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 initial horiz. velocity vix initial vertical velocity viy 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 = __________________ vix = __________________ ax = __________________ vfx = __________________ t = __________________ y = __________________ viy = __________________ 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 B focuses on horizontally launched projectile. 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 initial vertical velocity (viy) of 0 m/s. Any term containing the viy variable will thus cancel. 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 horizontal equations.

An expanded discussion of horizontally launched projectile problems is available at The Physics Classroom. The discussion includes several example problems with full solutions.

View Sample Problem Set.

 Problem Description Audio Link 1 A straight-forward 1-dimensional kinematic problem; solve for time using the set of vertical equations 2 A 2-dimensional equivalent of Problem 1; determine the time to fall from a given height 3 Extension of Problem 2; determine the horizontal displacement 4 Determination of the horizontal displacement using an x-y table 5 Extension of Problem 4; determine the final horizontal velocity; a conceptual brain-teaser 6 Extension of Problem 4; determine the final horizontal velocity 7 Determination of the vertical displacement from the given time 8 Extension of Problem 7; determine the horizontal displacement 9 Extension of Problem 7; determine the magnitude of the final velocity by combining vfx and vfy values using a vector diagram 10 Determination of the initial horizontal velocity using an x-y table 11 Determination of the vertical displacement using an x-y table 12 Determination of the initial horizontal velocity using an x-y table 13 Extension of Problem 12; determine the acceleration at the halfway point along the trajectory; a conceptual brain-teaser 14 Extension of Problem 12; determine the magnitude of the final velocity by combining vfx and vfy values using a vector diagram 15 Determination of the horizontal displacement using an x-y table 16 A complex physical situtation; determination of the horizontal displacement using an x-y table 17 Determination of the initial horizontal velocity using an x-y table

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