Physics 163

Unit 3: Motion in Two-Dimensions

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 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 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 viy value was equal to 0 m/s. Any expression with viy in it subsequently cancelled from the equation. This is not the case in Set D in which there is an initial component of velocity due to the angle launch. Thus, the equations are

The quantities vix and viy are the x- and y-components of the initial velocity. The values of vix and viy are related to the initial velocity (vi) and the angle of launch (). This relationship is depicted in the diagram and equations shown below.

The above two equations (affectionately known as the vixviy 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. For our purposes, we will be focus on projectiles which are launched from ground level, move 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 projectiles. 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 = viy + 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 = viy + ay • t

0 m/s = viy + (-9.8 m/s/s) • tup

tup = viy / (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 = viy

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 (vi) 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 vixviy 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.

An expanded discussion of non-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 Determine the total time if given vi and 2 Extension of Problem 1; determine the horizontal displacement 3 Extension of Problem 1; determine the peak height (vertical displacement at the halfway point of the trajectory) 4 Determine the horizontal displacement if given vi and 5 Extension of Problem 4; determine the peak height (vertical displacement at the halfway point of the trajectory) 6 Determine the total time if given vi and 7 Extension of Problem 6; determine the peak height (vertical displacement at the halfway point of the trajectory) 8 Extension of Problem 6; determine the horizontal displacement 9 Determine the initial velocity if given the angle and the ttotal 10 Extension of Problem 9; determine the peak height (vertical displacement at the halfway point of the trajectory) 11 A complex physical situation involving the motion of two objects

Audio Help for Problem: 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11

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