**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 v_{iy} value
was equal to 0 m/s. Any expression with v_{iy} 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 v_{ix} and
v_{iy} are the x- and y-components of the initial velocity.
The values of v_{ix} and v_{iy} are related to the
initial velocity (v_{i}) 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 v_{x} and v_{y} 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 v_{x} values remain constant throughout the trajectory.
The v_{y} 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 v_{y} 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
(t
_{total}) is thus the time up (t_{up}) to the peak multiplied by two:**t**_{total}= 2 • t_{up} - At the peak of the trajectory, there is no vertical velocity
for a projectile. The equation v
_{fy}= v_{iy}+ a_{y}• 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.v _{fy}= v_{iy}+ a_{y}• t0 m/s = v

_{iy}+ (-9.8 m/s/s) • t_{up}**t**_{up}= v_{iy}/ (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 (v_{i}) and the launch angle ().
The typical unknown quantities are:

- total time (t
_{total}) - horizontal displacement (x)
- vertical height above the ground at the peak
(y
_{peak})

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.

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