ChemPhys 173/273

Unit 7: Vectors and Projectiles

Problem Set D

Overview:

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

The Projectile Equations

The projectile equations introduced in Set C can be used in this Problem Set as well. However, in set C, the projectiles were launched horizontally and the voy value was equal to 0 m/s. Any expression with voy in it subsequently cancelled from the equation. This is not the case in Set D in which there is a horizontal component of velocity due to the angled launch. Thus, the equations are

Because the horizontal acceleration of a projectile is 0 m/s/s, the ax terms in the above equations are cancelled.

The quantities vox and voy are the x- and y-components of the original velocity. The values of vox and voy are related to the original velocity (vo) and the angle of launch (). This relationship is depicted in the diagram and equations shown below.

The above two equations (affectionately known as the voxvoy equations) are used on nearly every projectile problem in Problem Set D.

The Trajectory Diagram and Characteristics

Non-horizontally launched projectiles move horizontally above the ground as they move upward and downward through the air. One special case is a projectile which is launched from ground level, moves upwards towards a peak position, and subsequently fall from the peak position back to the ground. A trajectory diagram is often used to depict the motion of such a projectile. The diagram shown below depicts the path of the projectile and also displays the components of its velocity at regular time intervals.

The vx and vy vectors in the diagram represent the horizontal and vertical components of the velocity at each instant during the trajectory. A careful inspection shows that the vx values remain constant throughout the trajectory. The vy values decrease as the projectile rises from its initial location towards the peak position. As the projectile falls from its peak position back to the ground, the vy values increase. In other words, the projectile slows down as it rises upward and speeds up as it falls downward. This information is consistent with the definition of a projectile - an object whose motion is influenced solely by the force of gravity; such an object will experience a vertical acceleration only.

At least three other principles are observed in the trajectory diagram.

• The time for a projectile to rise to the peak is equal to the time for it to fall to the peak. The total time (ttotal) is thus the time up (tup) to the peak multiplied by two:
ttotal = 2 • tup
• At the peak of the trajectory, there is no vertical velocity for a projectile. The equation vfy = voy + ay • t can be applied to the first half of the trajectory of the projectile. In such a case, t represents tup and the vfy at this instant in time is 0 m/s. By substituting and re-arranging, the following derivation is performed.
vfy = voy + ay • t

0 m/s = voy + (-9.8 m/s/s) • tup

tup = voy / (9.8 m/s/s)

• The projectile strikes the ground with a vertical velocity which is equal in magnitude to the vertical velocity with which it left the ground. That is,
vfy = voy

These principles of projectile motion will need to be internalized and applied to the Set D word problems in order to be successful.

The Typical Problem

The typical non-horizontally launched projectile problem will include two pieces of explicitly stated information and include three requests for unknown information. The typical given values include the initial velocity (vo) and the launch angle (). The typical unknown quantities are:

1. total time (ttotal)
2. horizontal displacement (x)
3. vertical height above the ground at the peak (ypeak)

While not every problem in Set D fits into this mold, it is the case for problems 1-3 and 4-5 and 6-8. In such cases, you will need to make strong reliance upon the principles of a projectile's trajectory and the voxvoy equations in order to solve for the unknowns. The same principles will be used in the remainder of the problems in Set D, though the process will be less routine.

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. Some of the common 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 horizontal displacement (x) of an angled-launched projectile. 2 Referring to the previous problem; determination of the peak height (ypeak) of an angled-launched projectile. 3 Determination of the time of flight (t) of an angled-launched projectile. 4 Referring to the previous problem; determination of the peak height (ypeak) of an angled-launched projectile. 5 Referring to the previous problem; determination of the horizontal displacement (x) of an angled-launched projectile. 6 Determination of a building height (y) from knowledge of the launch speed and angle and the time of flight. 7 Referring to the previous problem; determination of the horizontal displacement (x) of an angled-launched projectile. 8 Determination of the launch velocity (vo) of a projectile from knowledge of the hang time and the angle of launch. 9 Referring to the previous problem; determination of the peak height (ypeak) of an angled-launched projectile. 10 Determination of the vertical displacement (y) of an angled-launched projectile from knowledge of the launch speed and angle and horizontal displacement. 11 Complex analysis comparing two thrown balls - one thrown vertically and the other thrown at an angle; must determine the speed at which the angled ball must be thown to reach the same height as the vertically-thrown ball. 12 Analysis of a horizontally-pitched baseball; must determine the wait time that a batter can maintain before swinging. 13 Determination of the launch speed of a trailbike in order to clear a ditch of known horizontal distance at a known angle of launch. 14 Determination of the launch speed of an angled launched projectile from knowledge of the launch angle and horizontal and vertical displacements. 15 A one-dimensional kinematic problem in which the speed of a car at the bottom of a steep hill is determined from knowledge of its acceleration and distance of travel. 16 Referring to the previous problem; determination of the time of flight of a downward-launched projectile from knowledge of the launch angle and speed and the distance of fall. 17 Referring to the previous problem; determination of the horizontal displacement (x) of a downward-launched projectile. 18 Determination of the vertical displacement (y) of an angled-launched projectile from knowledge of the launch angle and speed and horizontal displacement. 19 Determination of the launch speed of a motorcycle stuntman in order to clear a canyon of known horizontal distance at a known angle of launch. 20 Referring to the previous problem; determination of the peak height of the stuntman.

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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