GavinZhang Phys Project Part 1

7/29/2019 GavinZhang Phys Project Part 1

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Gavin Zhang

Rocket Projectile

Introduction

Displacement is the shortest distance of two points and it can express by the a

position vector's initial point and terminal point. Velocity is the speed with direction over

time. The different of the velocity and speed is speed only describe how fast the object

move and velocity also include the direction. Therefore velocity is a vector and speed is a

scalar. The velocity can be calculated by dividing the displacement with the time needed

for the displacement travelling. Acceleration is the rate of change in velocity, and it can

change the speed which is the magnitude of the vector, also the direction .

The projectile motion is created by shooting or throwing an object at a particular

angle and specific initial velocity. We can derive few kinematics equation from the

relationship between its position, velocity, acceleration, angle and time.

Variable is the horizontal position of any point of the projectile motion. isthe initial horizontal position of the projectile. can be found by adding the distance ofhorizontal travel by the projectile and it is the product of the horizontal velocity and time .


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Variable is the vertical position of any point of the projectile motion. is theinitial vertical position of the projectile. can be found by adding the distance of vertialtravel by the projectile and it is the product of the horizontal velocity and time .However, due to the gravity of the earth, we need to subtract the acceleration of gravity

because the direction is down by .

is the horizontal velocity at any point of the projectile motion. is the initialvelocity of the projectile, and we need to multiply it by to find out the horizontalcomponent of the vector. The direction of acceleration of gravity is vertical down, and it

won't affect the horizontal component.

is the vertical velocity at any point of the projectile motion. is the initialvelocity of the projectile, and we need to multiply it by to find out the verticalcomponent of the vector. The direction of acceleration of gravity is vertical down, and it

affect the vertical component over time by .The unit for position variables, and is . The unit for velocity variables

and is . The unit for acceleration variables is

Objective

For all those equation to work, we need to ignore some important factors in real

life and pretend they don't affect our calculation such as mass of the object, air friction,

wind, spinning of the object and the geography between two points.

Background

The application of projectile motion is widely used in daily life, sport and military.

The design of a drinking fountain may required some calculation of projectile, such as

how big the area of the faucet should be which is the maximum range at particular


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designed angle and the water pressure which is the initial velocity. After all those

calculation, we can get the best water pressure and angle that the water won't split out of

the faucet and best position for the person to drink the water [1]. According to

physicsweb.org [2], most sports have the action of throwing and jumping, and we can

find out the best angle of throwing using projectile motion in combine with knowledge of

biology study of muscles. Even the little boy playing around the classroom throwing

crumpled up paper into the trashcan involved projectile motion [3].

Method

The launching point of the rocket is Kalaupapa airport (Molokai, 211235N / 156 5823W) and the target point is Lanai City (Lanai, 20 4727N / 156 5717W), and

their respective elevations are 20.12m and 383.13m The map distance between two points

is 46.32km, which is 46,320m.


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These are the values known by obtaining from map and conversion,

= 0

= 0 = 46,320 = 383.13 - 20.12 = 363.01We know the following kinematic equation that respect to time (t),

Plot in the values we already know,

= 0 +


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= 0 + Solve for t with first equation,

Substitute t into second equation,

Solve for ,

=

Plot in

,

,

, and obtain

,

Results

We can start plotting in angles between 0 and 90 to find out the relationship

between initial angle and initial velocity. At first I tried some common angles such as 30, 45

and 60. However, I had tried more angles as well as the extreme cases, the 1 and 89 to get a

more accurate graph. After all the trials, table below is obtained:

*


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Angle/Velocity Combinations

Initial Angle

Initial Velocity

1 4858.528 m/s

5 1694.496 m/s

10 1178.539 m/s

20 849.5526 m/s30 728.9534 m/s

40 682.1175 m/s

45 676.4034 m/s

50 681.168 m/s

60 725.6325 m/s

70 841.5576 m/s

80 1152.847 m/s

85 1617.377 m/s

89 3606.762 m/s


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Use these data and plot them into an xy coordinated graph, following graph is

obtained:

1694.50

1178.54

849.55

728.95

682.12

676.40

681.17

725.63

841.56

1152.85

1617.38

600.00

700.00

800.00

900.00

1000.00

1100.00

1200.00

1300.00

1400.00

1500.00

1600.00

1700.00

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Velocity

(m/s)

Angle ()

Initial Velocity vs Initial Angle


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A simple graph of simulated projectile motion:

Combination of angle and velocity of my choice:


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Time to reach maximum height:

*The time can obtain by setting ,

*Solve for ,

Maximum height to coordinate system:

*Provide the y position equation,

*Plot in all the known values,

*Solve for ,

Maximum height to sea level:

*Add up the maximum height to the coordinate system and the height of the

coordinate system to the sea level,


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Time to reach target:

*Provide x position equation,

*Plot in all the known values,

*Solve for ,

Final velocity right before impact:

*Final velocity need to calculate with both horizontal and vertical components

*Provide horizontal velocity equation,

*Solve for

*Provide vertical velocity equation,

*Solve for


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*Solve for

Table of results with combination of 728.9524 m/s and 30

Variable Description Value

Final horizontal position 46320 m Final vertical position 363.01 m

Maximum vertical position 6777.698988 m Initial horizontal position 0 m Initial vertical position 0 m Initial velocity 728.9524 m/s Initial angle 30

Time at maximum height 37.19 s Time at final position 73.37 s

Final horizontal velocity 631.29 m/s Final vertical velocity -354.55 m/s Final velocity 727.04 m/s


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Analysis

The result data shows us that we can accurately calculate every expects within the

projectile motion system using the equations under the assumptions of neglecting the air

friction and the spinning effect of the object. If we count the air friction in, it will cause

an acceleration with a direction opposite to the direction of motion. Therefore, the object

will won't go as high as before and it won't accurately hit the target. We need to measure

the extra distance and increase initial velocity to fulfill the condition. However there are

more condition in the nature such as wind will affect the projectile motion, we need to

consider all of those factors when calculating.

Conclusion

Under the condition of neglecting all those complicate factors, air friction,

spinning and wind, we can calculate the projectile motion between Kalaupapa airport and

Lanai City using basic physics knowledge of projectile. We can find out the initial

velocity needed for their related angle, and it is

References:

[1] Water Coolers and Water Fountains,http://www.elkayusa.com/Files_Media/F-

4293%20Water%20Cooler%20Catalog%20DFC-19.pdf

[2] Physicsweb.org,http://images.iop.org/dl/physicsweb/PWJUNE06linthorne.pdf

[3] Projectile MotionSeptember 9, 2011,

http://tiffyu.wordpress.com/2011/09/10/projectile-motion-september-9-2011/
http://www.elkayusa.com/Files_Media/F-4293%20Water%20Cooler%20Catalog%20DFC-19.pdfhttp://www.elkayusa.com/Files_Media/F-4293%20Water%20Cooler%20Catalog%20DFC-19.pdfhttp://www.elkayusa.com/Files_Media/F-4293%20Water%20Cooler%20Catalog%20DFC-19.pdfhttp://www.elkayusa.com/Files_Media/F-4293%20Water%20Cooler%20Catalog%20DFC-19.pdfhttp://images.iop.org/dl/physicsweb/PWJUNE06linthorne.pdfhttp://images.iop.org/dl/physicsweb/PWJUNE06linthorne.pdfhttp://images.iop.org/dl/physicsweb/PWJUNE06linthorne.pdfhttp://tiffyu.wordpress.com/2011/09/10/projectile-motion-september-9-2011/http://tiffyu.wordpress.com/2011/09/10/projectile-motion-september-9-2011/http://tiffyu.wordpress.com/2011/09/10/projectile-motion-september-9-2011/http://images.iop.org/dl/physicsweb/PWJUNE06linthorne.pdfhttp://www.elkayusa.com/Files_Media/F-4293%20Water%20Cooler%20Catalog%20DFC-19.pdfhttp://www.elkayusa.com/Files_Media/F-4293%20Water%20Cooler%20Catalog%20DFC-19.pdf

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GavinZhang Phys Project Part 1