Does the kinematic motion of an object depend on the object’s mass or other properties of the object? Why or Why Not?
PHYS 103
Report 2 Instructions
Projectile Motion
Download this document and record the results in the table below as prompted by the procedure. Save the completed document to your computer, include photos, and upload the final 1–2-page Word document.
Take a photo of the setup at the prompt in the assignment.
The Data Table is intended to be filled out from left to right with the columns before calculated distance being computed by the principles within the laboratory exercise. Using the data table, calculate for an initial angle (and incremented angles thereafter), the acceleration of the steel ball, the velocity, time until the ball hits the ground, and the horizontal distance the ball should travel. You will mark the calculated position on the floor and then measure the actual distance traveled.
Submit photo of completed table showing all your work.
2.The laboratory exercise for this chapter addresses kinematic motion. You have experienced these motions in everyday life.
Discuss the differences between speed and velocity own your word.
Does the kinematic motion of an object depend on the object’s mass or other properties of the object? Why or Why Not?
Discuss 1 or 2 examples of kinematic motion that you may have encountered.
Submit Report 2 by 11:59 p.m. (ET) on Monday of Module/Week 3.
Projectile Motion
Carolina Distance Learning
Investigation Manual
2
©2015 Carolina Biological Supply Company
Table of Contents
Overview …………………………………………………………………………………………. 3
Objectives ……………………………………………………………………………………….. 3
Time Requirements ………………………………………………………………………….. 3
Background …………………………………………………………………………………….. 4
Materials ………………………………………………………………………………………….. 7
Safety ………………………………………………………………………………………………. 8
Preparation ……………………………………………………………………………………… 8
Activity 1: Projectile launched in a horizontal direction. ………………. 10
Data Table …………………………………………………………………………………….. 12
3
©2015 Carolina Biological Supply Company
Overview
In this investigation, students will explore two-dimensional motion and learn how
vectors are used to describe the trajectory of an object. Students will observe the
motion of an object launched horizontally at various speeds, and they will learn how
to predict the motion of the launched object using their prior knowledge of kinematics
combined with new knowledge of vectors and the trajectory of projectiles.
Objectives
Describe what factors affect the trajectory of a projectile
Explain how vectors are used to describe two-dimensional and projectile motion
Predict the trajectory of a horizontally launched projectile using vectors and
kinematics equations.
Time Requirements
Preparation …………………………………………………………………………………15 minutes
Activity 1 …………………………………………………………………………………….30 minutes
Activity 2 …………………………………………………………………………………….20 minutes
4
©2015 Carolina Biological Supply Company
Background
Projectiles are objects that are given an initial velocity and subsequently travel along
their trajectory (flight path) due to their own inertia. Vectors describe the velocity,
acceleration, and forces that act upon a projectile in terms of direction and
magnitude. The principles of vector addition are used to understand and predict the
trajectory of projectiles and can be used in other applications of two-dimensional
motion, such as circular motion or the elliptical orbits of planets and comets.
Therefore, vector addition is an important subject in the field of mechanics, a branch
of physics that studies how physical bodies behave when subjected to forces or
displacements.
Once the motion of a projectile is understood, knowledge of a few initial parameters
can allow the calculation of many aspects of the projectile’s trajectory, such as the
maximum height, the time of flight, and the range, or the horizontal distance the
object will travel.
A simple example of a projectile is a ball that is thrown. A ball thrown with less force
has a lower speed and hits the ground sooner and nearer than the same ball thrown
with greater force. However, the angle at which the ball is thrown also affects the
trajectory of the ball. Which matters more, the initial speed or the release angle?
What happens when a ball is thrown at a high speed, but at a shallow angle? Will it
travel farther than a ball traveling at a low speed at a greater angle? The answers to
these questions can all be calculated by applying kinematics equations and some
knowledge about vectors.
Projectiles will tend to follow a parabolic trajectory. If you draw a line that follows the
movement of a ball after you throw it, you would see the shape of a parabola. The
shape of the parabola depends on the initial speed and the release angle, but all
projectiles launched at an angle follow this parabolic curve (see Figure 1).
Figure 1.
5
©2015 Carolina Biological Supply Company
In order to understand the motion of a projectile, it helps to consider the object as
moving in two dimensions, the vertical (y) direction and the horizontal (x) direction.
The velocity of the projectile at any given time can be broken down or resolved into a
vector in the x direction and a vector in the y direction. The magnitudes of these
vectors are independent of one another. Gravity only affects the vertical component
of the velocity, not the horizontal component.
Consider Figure 1. When the projectile is launched, the velocity, v, consists of two
independent, perpendicular components, vx, and vy. If air resistance is negligible, the
horizontal component of the velocity, vx, remains constant, whereas the vertical
component of the velocity vy changes due to gravitational acceleration. The initial
value for vy decreases as the projectile travels to the highest point in the parabolic arc
and then increases in the opposite direction as the projectile descends. If air
resistance is negligible, the vertical velocity of the projectile when it returns to the
elevation from which it was launched will have the same magnitude as when the
projectile was launched, but the direction will have turned 180°.
Consider two projectiles launched horizontally at exactly the same time and from the
same height, but one projectile has an initial velocity that is twice the other projectile.
If the ground beneath the projectiles is level and air resistance is ignored, both
projectiles will land on the ground at the same time. This may seem counter intuitive,
because the projectile with the greater speed is traveling farther, but experimentation
proves that the time of flight of both projectiles will be the same, and both projectiles
will land at the same time. The projectile with the greater velocity will land farther and
its parabolic trajectory will be different, but the time for the two projectiles to reach
the ground is the same.
When air resistance is taken into account, the mathematics describing the motion of
projectiles can be challenging, but in many cases the air resistance is negligible and
can be ignored. If air resistance is ignored, the motion of a projectile can be
described by kinematics equations. The motion in the horizontal direction is constant
and can be described with this simple equation:
=
where vx is the magnitude of the horizontal component of the projectile’s velocity, x is
the horizontal distance that the object travels, and t is the time. Although the
projectile’s velocity in the horizontal direction is constant, its velocity in the y direction
is constantly being accelerated by gravity at a rate of g = 9.8 m/s2.
If a projectile is fired at an angle of 0° from the horizontal (i.e. in the x direction), the time
for the projectile to fall to the ground depends only on the initial height and the
acceleration due to gravity. The time is independent of the horizontal velocity.
The motion of the projectile in the y direction, which is affected due to the acceleration
of gravity, can be described by kinematics equations, as follows:
6
©2015 Carolina Biological Supply Company
= 1
2 2 +
2 2 = 1
2 + 2
2 = 1 +
= 1
2 ( + )
where y is the displacement of the projectile in the y direction, a is the acceleration in
the y direction (which in this context is equal to the acceleration due to gravity, g = 9.8
m/s2), vy2 is the velocity of the object in the y direction at time t2, vy1 is the velocity in the
y direction at time t1, and t is the time of flight between t1 and t2.
Because the magnitudes of perpendicular vectors are independent of each other, the
time that a projectile travels can be calculated by considering only the vertical
component of the velocity. Once the time of flight for the projectile is known, the
horizontal distance that the object travels can be calculated by multiplying this time by
the horizontal speed of the projectile.
In this activity, you will predict and then measure the horizontal distance of a projectile
launched from an elevated position with an initial velocity that has only a horizontal
component. In order to measure the horizontal distance that the projectile will travel,
you will need to know the horizontal speed of the projectile, and the time that the
projectile will be in the air.
The projectile in this activity will be the steel sphere from the mechanics materials kit.
The sphere will roll down an incline using the angle bar as a track, then transition to a
grooved ruler so that it will travel horizontally when it leaves the table. You will apply
your knowledge of kinematics to determine the velocity of the sphere as it leaves the
table.
Since the sphere has no vertical velocity as it leaves the table, the time for the sphere
to reach the ground is determined only by the height of the table and the
acceleration due to gravity, which will be g = 9.8 m/s2.
7
©2015 Carolina Biological Supply Company
Materials
Included in the Mechanics Module kit:
Metal Sphere 1
Acrylic Sphere 1
Angle Bar 1
Clay 1
Needed from the Central Materials set:
Ruler 1
String 1
Washer 1
Tape Measure 1
Protractor 1
Needed, but not supplied:
Book 1
Masking Tape 1
Calculator 1
Reorder Information: Replacement supplies for the Projectile Motion investigation can
be ordered from Carolina Biological Supply Company, Conceptual Physics Mechanics
Module kit 580404.
Call 1-800-334-5551 to order.
8
©2015 Carolina Biological Supply Company
Safety
Safety Goggles should be worn at all times during this
experiment.
Read all the instructions for this laboratory activity before beginning. Follow the
instructions closely and observe established laboratory safety practices, including
the use of appropriate personal protective equipment (PPE) as described in the
Safety and Procedure section.
Safety Goggles should be worn during this experiment.
Do not eat, drink, or chew gum while performing this activity. Wash your hands with
soap and water before and after performing the activity. Clean up the work area
with soap and water after completing the investigation. Keep pets and children
away from lab materials and equipment.
Preparation
1. Locate a smooth, level surface or table at least 70 cm from the floor.
2. Clear the table and the floor in front of the table.
3. Place the book on the table so that one end of the angle bar may rest on the
book and the other end stops about 5 centimeters from the end of the table
(see Figure 2).
4. Place some clay on the book to create a seat for the angle bar.
5. Place the grooved ruler at the end of the angle bar so that the angle bar rests in
the groove of the ruler, and the ruler runs to the end of the table.
6. Tape the yellow ruler to the table to keep the ruler in place. Place the tape
behind the point where the angle bar rests on the ruler so that the tape does not
interfere with the sphere as it rolls.
Note: For this experiment the sphere must roll down the angle bar and leave
the table with a horizontal velocity. The groove in the ruler allows the
sphere to transition from the incline to a horizontal direction.
9
©2015 Carolina Biological Supply Company
7. On the edge of the table, just below the end of the ruler, tape a piece of string
and allow it to hang vertically from the table. The string should stop about 3 cm
from the floor.
8. At the bottom end of the string, tie a washer. This is a plumb line, and it will allow
you to find the point on the floor directly below the point where the sphere will
leave the table.
Measure the angle of the angle bar vs the table with the protractor (see Figure
2). Record the value in the Data Table in the column titled θ for Trial 1.
9. Mark a point about 3 cm from the higher end of the angle bar. This will be the
start point.
10. Take a photograph of your complete setup.
Figure 2
Book
10
©2015 Carolina Biological Supply Company
Activity 1: Projectile launched in a horizontal direction.
1. Measure straight down from the top of the table to the floor with the tape
measure. Follow the plumb line to make sure the tape measure is straight.
2. Rearrange the kinematics equation for vertical displacement s to give an
equation for time. Calculate the value for time using this equation and write it in
the Data Table. This time should be the same for each trial.
= 1
2 ∆ 2
In other words,
= √ 2
Because the sphere is in free-fall after it leaves the table, the acceleration will be
equal to gravitational acceleration:
= = 9.8
2
The displacement s is the vertical height from the table to the floor.
= ℎ
Therefore, the equation for the time of flight, t, can be rewritten as
= √ 2ℎ
3. Calculate the horizontal velocity the sphere will have as it leaves the table by
calculating the velocity of the sphere at the bottom of the incline.
First calculate the acceleration of the sphere as it rolls down the incline. The
acceleration of the sphere as it rolls is given by:
= 0.7 sin
substitute the angle of the incline for θ, and record the value for acceleration in
the Data Table.
4. Use this value for the acceleration to find the horizontal speed of the sphere as it
leaves the table, by applying the following kinematics equation:
11
©2015 Carolina Biological Supply Company
2 = 1
2 + 2
where vx is the translational velocity of the sphere as it reaches the bottom of the
track, v1 is the initial velocity of the sphere, and is 0 m/s, because the marble will
be released from rest. a is the acceleration of the sphere. s is the length of the
track from the start point to the end of the slope. Assume the sphere travels at
this speed along the length of the horizontal ruler.
Rearrange the equation and substitute the value for a from the previous
calculation, and the length of the angle bar from the start point to the end of
the ramp.
= √2
Record the value for vx in the Data Table.
5. Multiply the value for the horizontal velocity, vx by the time found in step 2.
Record the value (in meters) in the Data Table.
This is the distance that the sphere will travel before it strikes the floor.
6. Using the tape measure, find the point on the floor that is at the same distance
from the table as the value calculated in step 5. Measure from directly beneath
the plumb line, and measure in the same direction that the angle bar is pointing.
7. Place the sphere at the start point on the high end of the angle bar.
8. Release the sphere, and allow the sphere to roll down the angle bar, to the
grooved ruler, and off the table. The sphere should land on or close to the point
you marked on the floor.
9. Measure the distance to the point where the steel sphere struck the floor.
10. Find the percent error between the distance you calculated and the distance
actually traveled by the sphere.
= | − |
100%
11. Repeat the experiment with the acrylic sphere.
12. Repeat the experiment using both the steel and acrylic spheres, increasing the
angle by 5° then 10°.
12
©2015 Carolina Biological Supply Company
Data Table
Data Table
Trial Sphere θ a = 0.71(9.8)sinθ
= √2 = √
2ℎ
Calculated
Distance =
Actual
Distance
Percent
Difference
1 Steel
2 Steel +5°
3 Steel +10°
4 Acrylic
5 Acrylic +5°
6 Acrylic +10°