Homework for lab 10 two dimensional motion answers

Projectile Motion

Carolina Distance Learning

Investigation Manual

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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

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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

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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.

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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:

𝑣𝑥 = 𝑥

𝑡

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:

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𝑦 = 1

2 𝒂𝑡2 + 𝒗𝒚𝟏𝛥𝑡

𝒗𝑦2 2 = 𝒗𝑦1

2 + 2𝒂𝒚

𝒗𝑦2 = 𝒗𝑦1 + 𝒂𝛥𝑡

𝑦 = 1

2 (𝒗𝒚𝟏 + 𝒗𝒚𝟐)𝛥

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.

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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.

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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.

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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

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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.

𝒔 = 1

2 𝒂∆𝑡2

In other words,

𝑡 = √ 2𝑠

𝒂

Because the sphere is in free-fall after it leaves the table, the acceleration will be

equal to gravitational acceleration:

𝒂 = 𝒈 = 9.8 𝑚

𝑠2

The displacement s is the vertical height from the table to the floor.

𝒔 = ℎ

Therefore, the equation for the time of flight, t, can be rewritten as

𝑡 = √ 2ℎ

𝒈

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:

𝒂 = 0.7𝒈 sin 𝜃

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:

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𝒗𝑥 2 = 𝒗1

2 + 2𝒂𝒔

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.

𝑣𝑥 = √2𝑎𝑠

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.

𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟 = |𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 − 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙|

𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑥 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°.

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Data Table

Data Table

Trial Sphere θ a = 0.71(9.8)sinθ

𝒗𝒙 = √2𝒂𝒔 𝑡 = √

2ℎ

𝒈

Calculated

Distance 𝑥 = 𝒗𝒙𝑡

Actual

Distance

Percent

Difference

1 Steel

2 Steel +5°

3 Steel +10°

4 Acrylic