Objective Theory

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Lab Manual

General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Free Fall & Projectile Motion

Ver.20180302

[International Campus]

Free Fall and Projectile Motion

Objective

Investigate the motions of a freely falling body and a projectile under the influence of gravity. Find the acceleration due to gravity.

Theory

-----------------------------

Reference

-------------------------th

Young & Freedman, University Physics (14 ed.), Pearson, 2016

Figure 2 is the π‘₯π‘₯-𝑑𝑑 graph of the car’s position as a function

of time. The average velocity of the car equals the slope of the line 𝑝𝑝1 𝑝𝑝2. But the average velocity during a time interval

2.1 Displacement, Time, and Average Velocity (p.58~61)

can’t tell us how fast, or in what direction. To do this we need

2.2 Instantaneous Velocity (p.61~64)

to know the instantaneous velocity, or the velocity at a spe-

2.3 Average and Instantaneous Acceleration (p.64~68)

cific instant of time or specific point along the path.

2.4 Motion with Constant Acceleration (p.69~74) 2.5 Freely Falling Bodies (p.74~77)

The instantaneous velocity is the limit of the average veloci-

3.3 Projectile Motion (p.99~106) -----------------------------------------------------------------------------

ty as the time interval approaches zero. On the π‘₯π‘₯-𝑑𝑑 graph

(Fig. 2), the instantaneous velocity at any point is equal to the When a car moves from 𝑃𝑃1 to 𝑃𝑃2 in the +π‘₯π‘₯-direction as in

slope of the tangent to the curve at that point.

figure 1, the π‘₯π‘₯ -component of average velocity is the π‘₯π‘₯ -

𝑣𝑣π‘₯π‘₯ = lim

component of displacement βˆ†π‘₯π‘₯ = π‘₯π‘₯2 βˆ’ π‘₯π‘₯1 divided by the time

βˆ†π‘‘π‘‘β†’0

interval βˆ†π‘‘π‘‘ = 𝑑𝑑2 βˆ’ 𝑑𝑑1 during which the displacement occurs. 𝑣𝑣av-π‘₯π‘₯ =

Fig. 1

π‘₯π‘₯2 βˆ’ π‘₯π‘₯1 βˆ†π‘₯π‘₯ = 𝑑𝑑2 βˆ’ 𝑑𝑑1 βˆ†π‘‘π‘‘

Positions of a car at two times during its run.

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βˆ†π‘₯π‘₯ 𝑑𝑑𝑑𝑑 = βˆ†π‘‘π‘‘ 𝑑𝑑𝑑𝑑

(2)

(1)

Fig. 2

The position of a car as a function of time.

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Lab Manual

General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Free Fall & Projectile Motion

Ver.20180302

Acceleration describes the rate of change of velocity with

We can also derive an equation for the position π‘₯π‘₯ as a func-

time. Suppose that at time 𝑑𝑑1 the object is at point 𝑃𝑃1 and

tion of time using Eqs. (1) and (5) when the π‘₯π‘₯-acceleration is

at point 𝑃𝑃2 and has velocity 𝑣𝑣2π‘₯π‘₯ . So the velocity changes by

position π‘₯π‘₯ at the later time 𝑑𝑑, Eq. (1) becomes

has π‘₯π‘₯-component of velocity 𝑣𝑣1π‘₯π‘₯ , and at a later time 𝑑𝑑2 it is

amount βˆ†π‘£π‘£π‘₯π‘₯ = 𝑣𝑣2π‘₯π‘₯ βˆ’ 𝑣𝑣1π‘₯π‘₯ during βˆ†π‘‘π‘‘ = 𝑑𝑑2 βˆ’ 𝑑𝑑1 . We define the

constant. With the initial position π‘₯π‘₯0 at time 𝑑𝑑 = 0 and the

average acceleration of the object equals βˆ†π‘£π‘£π‘₯π‘₯ divided by

βˆ†π‘‘π‘‘.

π‘Žπ‘Žav-π‘₯π‘₯ =

𝑣𝑣2π‘₯π‘₯ βˆ’ 𝑣𝑣1π‘₯π‘₯ βˆ†π‘£π‘£π‘₯π‘₯ = 𝑑𝑑2 βˆ’ 𝑑𝑑1 βˆ†π‘‘π‘‘

(3)

We can now define instantaneous acceleration following

𝑣𝑣av-π‘₯π‘₯ =

π‘₯π‘₯ βˆ’ π‘₯π‘₯0 𝑑𝑑

(6)

We can also get a second expression for 𝑣𝑣avΛ—Μ΅π‘₯π‘₯ . In this case

the average π‘₯π‘₯-velocity for the time interval from 0 to 𝑑𝑑 is simply the average of 𝑣𝑣0π‘₯π‘₯ and 𝑣𝑣π‘₯π‘₯ .

the same procedure that we used to define instantaneous

𝑣𝑣avΛ—Μ΅π‘₯π‘₯ =

velocity. The instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero.

𝑣𝑣0π‘₯π‘₯ + 𝑣𝑣π‘₯π‘₯ 2

(7)

Substituting Eq. (5) into Eq. (7) yields π‘Žπ‘Žπ‘₯π‘₯ = lim

βˆ†π‘‘π‘‘β†’0

βˆ†π‘£π‘£π‘₯π‘₯ 𝑑𝑑𝑣𝑣π‘₯π‘₯ = βˆ†π‘‘π‘‘ 𝑑𝑑𝑑𝑑

(4)

The simplest kind of accelerated motion is straight-line mo-

1 1 𝑣𝑣avΛ—Μ΅π‘₯π‘₯ = (𝑣𝑣0π‘₯π‘₯ + 𝑣𝑣0π‘₯π‘₯ + π‘Žπ‘Žπ‘₯π‘₯ 𝑑𝑑) = 𝑣𝑣0π‘₯π‘₯ + π‘Žπ‘Žπ‘₯π‘₯ 𝑑𝑑 2 2

We set Eq. (6) and Eq. (8) equal to each other and simplify

tion with constant acceleration. We can find the velocity 𝑣𝑣π‘₯π‘₯ of

1 π‘₯π‘₯ = π‘₯π‘₯0 + 𝑣𝑣0π‘₯π‘₯ 𝑑𝑑 + π‘Žπ‘Žπ‘₯π‘₯ 𝑑𝑑 2 2

that motion using Eq. (3). We use 𝑣𝑣0π‘₯π‘₯ for the π‘₯π‘₯-velocity at

𝑑𝑑 = 0; the π‘₯π‘₯-velocity at the later time 𝑑𝑑 is 𝑣𝑣π‘₯π‘₯ . Then Eq. (3)

becomes

𝑣𝑣π‘₯π‘₯ βˆ’ 𝑣𝑣0π‘₯π‘₯ π‘Žπ‘Žπ‘₯π‘₯ = 𝑑𝑑 βˆ’ 0

𝑣𝑣π‘₯π‘₯ = 𝑣𝑣0π‘₯π‘₯ + π‘Žπ‘Žπ‘₯π‘₯ 𝑑𝑑

or

(8)

(9)

Figure 4 shows the graphs of Eq. (9) and Eq. (5). If there is zero π‘₯π‘₯-acceleration, the π‘₯π‘₯-𝑑𝑑 graph is a straight line; if there

(5)

is a constant π‘₯π‘₯-acceleration, the additional (1⁄2)π‘Žπ‘Žπ‘₯π‘₯ 𝑑𝑑 2 term

curves the graph into a parabola (Fig. 4(a)). Also, if there is zero π‘₯π‘₯-acceleration, the 𝑣𝑣π‘₯π‘₯ -𝑑𝑑 graph is a horizontal line; add-

ing a constant π‘₯π‘₯-acceleration gives a slope to the 𝑣𝑣π‘₯π‘₯ -𝑑𝑑 graph

(Fig. 4(b)).

Fig. 3

A 𝑣𝑣π‘₯π‘₯ -𝑑𝑑 graph for an object moving with constant acceleration on π‘₯π‘₯-axis.

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Fig. 4

How a constant π‘₯π‘₯-acceleration affects a body’s (a) π‘₯π‘₯-𝑑𝑑 graph and (b) 𝑣𝑣π‘₯π‘₯ -𝑑𝑑 graph.

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The most familiar example of motion with constant acceleration is a body falling under the influence of the earth’s gravita-

Free Fall & Projectile Motion

Ver.20180302

We can then express all the vector relationships for the position, velocity, and acceleration by separate equations

tional attraction. If the distance of the fall is small compared π‘Žπ‘Žπ‘₯π‘₯ = 0

with the radius of the earth, and if the effects of the air can be neglected, all bodies fall with the same downward accelera-

π‘Žπ‘Žπ‘¦π‘¦ = βˆ’π˜¨π˜¨

(10)

tion. This is called free fall. Fig. 5 shows successive images

Since both are constant, we can use Eqs. (5) and (9) directly.

of falling bodies separated by equal time intervals. The red

For example, as in Fig. 6, suppose that at time 𝑑𝑑 = 0 our

ball is dropped from rest. There are equal time intervals between images, so the average velocity of the ball between successive images is proportional to the distance between them. The increasing distances between images show that the velocity is continuously changing. Careful measurement shows that the acceleration of the freely falling ball is constant. This acceleration is called the acceleration due to gravity. We denote its magnitude with 𝘨𝘨. The approximate

projectile is at the point (π‘₯π‘₯0 , 𝑦𝑦0 ) = (0, 0) and that at this time

its velocity components have the initial values 𝑣𝑣0π‘₯π‘₯ = 𝑣𝑣0 cos 𝛼𝛼0 and 𝑣𝑣0𝑦𝑦 = 𝑣𝑣0 sin 𝛼𝛼0 . From Eqs. (5), (9) and (10), we find 𝑣𝑣π‘₯π‘₯ = 𝑣𝑣0π‘₯π‘₯ + π‘Žπ‘Žπ‘₯π‘₯ 𝑑𝑑 = 𝑣𝑣0 cos 𝛼𝛼0

𝑣𝑣𝑦𝑦 = 𝑣𝑣0𝑦𝑦 + π‘Žπ‘Žπ‘¦π‘¦ 𝑑𝑑 = 𝑣𝑣0 sin 𝛼𝛼0 βˆ’ 𝘨𝘨𝘨𝘨

glect the effects of air resistance.) The motion of the yellow ball in Fig. 5 is two-dimensional. We will call the plane of motion the π‘₯π‘₯π‘₯π‘₯-coordinate plane, with the π‘₯π‘₯-axis horizontal and the 𝑦𝑦-axis vertically upward. The π‘₯π‘₯-component of accelera-

tion is zero, and the 𝑦𝑦-component is constant and equal to

βˆ’π˜¨π˜¨. So we can analyze projectile motion as a combination of

horizontal motion with constant velocity and vertical motion with constant acceleration.

(13)

1 1 𝑦𝑦 = 𝑦𝑦0 + 𝑣𝑣0𝑦𝑦 𝑑𝑑 + π‘Žπ‘Žπ‘¦π‘¦ 𝑑𝑑 2 = (𝑣𝑣0 sin 𝛼𝛼0 )𝑑𝑑 βˆ’ π˜¨π˜¨π‘‘π‘‘ 2 2 2

A projectile, such as a thrown baseball, is any body that is

entirely by the effects of gravitational acceleration. (We ne-

(12)

1 π‘₯π‘₯ = π‘₯π‘₯0 + 𝑣𝑣0π‘₯π‘₯ 𝑑𝑑 + π‘Žπ‘Žπ‘₯π‘₯ 𝑑𝑑 2 = (𝑣𝑣0 cos 𝛼𝛼0 )𝑑𝑑 2

value of 𝘨𝘨 near the earth’s surface is 9.8 m⁄s2 .

given an initial velocity and then follows a path determined

(11)

(14)

The time 𝑑𝑑1 when the projectile hits the ground is 1 0 = (𝑣𝑣0 sin 𝛼𝛼0 )𝑑𝑑1 βˆ’ π˜¨π˜¨π‘‘π‘‘12 2

or

𝑑𝑑1 =

2𝑣𝑣0 sin 𝛼𝛼0 𝘨𝘨

(15)

The horizontal range 𝑅𝑅 is the value of π‘₯π‘₯ at this time. Sub-

stituting equation (15) into equation (13) yields

𝑅𝑅 = (𝑣𝑣0 cos 𝛼𝛼0 )𝑑𝑑1 =

2𝑣𝑣02 sin 𝛼𝛼0 cos 𝛼𝛼0 𝑣𝑣02 = sin 2𝛼𝛼0 𝘨𝘨 𝘨𝘨

(16)

In Eq. (16), the maximum value of sin 2𝛼𝛼0 is 1. This occurs

when 2𝛼𝛼0 = 90Β° or 𝛼𝛼0 = 45Β°. This angle gives the maximum

𝑅𝑅 for a given initial speed if air resistance can be neglected.

Fig. 5

The red ball is dropped from rest, and the yellow ball is simultaneously projected horizontally; successive images in this stroboscopic photograph are separated by equal time intervals. At any given time, both balls have the same 𝑦𝑦-position, 𝑦𝑦-velocity, and 𝑦𝑦-acceleration, despite having different π‘₯π‘₯-positions and π‘₯π‘₯-velocities.

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Fig. 6

The trajectory of a projectile is a combination of horizontal motion with constant velocity and vertical motion with constant acceleration.

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General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Free Fall & Projectile Motion

Ver.20180302

Equipment

1. List Item(s)

PC / Software Data Analysis: Capstone

Qty.

1

Description

Records, displays and analyzes data measured by various sensors.

Data acquisition interface designed for use with various Interface

1

sensors, including power supplies which provide up to 15 watts of power.

Photogate (Rod, Cable, and Screw included)

1 set

Measures high-speed or short-duration events.

A-shaped Base

1

Multi-clamp

1

Support Rod (600mm)

1

Provides stable support for experiment set-ups.

Cushioned Baskets

1

Absorb shock on impact.

Picket Fence set

1 set

Provide stable support for experiment set-ups.

PF#1: The edges of the bands are 50mm apart. (Opaque: 20mm / Transparent: 30mm)

PF#2: The edges of the bands are 40mm apart. (Opaque: 20mm / Transparent: 20mm)

Projectile Launcher

1

Photogate Bracket

1

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Launches a ball at any angle from zero to ninety degrees with three range settings.

Mounts the Photogate on the Projectile Launcher.

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Item(s)

Projectile (Green Plastic Ball)

Free Fall & Projectile Motion

Ver.20180302

Qty.

1

Description

Green plastic ball which can be loaded into the Projectile Launcher.

Table Clamp

1

Clamps the Projectile Launcher to a lab table.

Carbon Paper

1

Leaves a mark when a projectile ball hits it.

White Paper

1

(White Paper is not provided.)

Provides a horizontal surface so the projectile ball can Box

1

reaches the same level as the muzzle of the Projectile Launcher.

Measuring Tape

1

Vernier Caliper

1

Measures distance.

Measures external, internal diameter or depth of an object with a precision to 0.05mm.

2. Details (1) Interface

(2) Capstone: Data Acquisition and Analysis Software

The Capstone software records, displays and analyzes data The 850 Universal Interface is a data acquisition interface designed for use with various sensors to measure physical

measured by the sensor connected to the 850 interface. It also controls the built-in signal generator of the interface.

quantities; position, velocity, acceleration, force, pressure, magnetic field, voltage, current, light intensity, temperature, etc. It also has built-in signal generator/power outputs which provide up to 15 watts DC or AC in a variety of waveforms such as sine, square, sawtooth, etc. Lab Office (Int’l Campus)

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Lab Manual

General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Free Fall & Projectile Motion

Ver.20180302

(3) Photogate

The Photogate sensor is an optical timing device used for very precise measurements of high-speed or short-duration events. It consists of a light source (infrared LED) and a light detector (photodiode). When an object moves through and blocks the infrared beam between the source and the detec-

The Projectile Launcher has three range settings so that

tor, a signal is produced which can be detected by the inter-

balls can be launched with three different initial speeds. One

face.

or two Photogates can be attached to the Projectile Launcher using the Photogate Bracket so that the photogates can

When the infrared beam is blocked, the output signal of the

measure the initial speed of the ball.

photogate becomes β€˜0’ and the LED lamp on the photogate goes on. When the beam is not blocked, the output signal

(5) Vernier Caliper

becomes β€˜1’ and the LED goes off. This transition of signal can be used to calculate quantities such as the period of a pendulum, the velocity of an object, etc.

(4) Projectile Launcher

The Projectile Launcher is designed for projectile motion experiments. Balls can be launched from any angle from zero to ninety degrees measured from horizontal. The angle is easily adjusted using thumbscrews, and the built-in protractor and plumb-bob give and accurate way to measure the angle of inclination.

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β‘  22 mm is to the immediate left of the zero on the vernier scale. Hence, the main scale reading is 22 mm.

β‘‘ Look closely for and alignment of the scale lines of the th

main scale and vernier scale. In the figure, the aligned (13 ) line corresponds to 0.65 mm (= 0.05 Γ— 13).

β‘’ The final measurement is given by the sum of the two readings. This gives 22.65 mm (= 22 + 0.65).

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Lab Manual

General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Free Fall & Projectile Motion

Ver.20180302

Procedure

Experiment 1. Free Fall

The interface is automatically detected by Capstone. Click [Hardware Setup] in the [Tools] palette to configure the inter-

(1) Set up equipment as below.

face.

(3) Set up Capstone software

(3-1) Add a Photogate.

Click the input port which you plugged the Photogate into and select [Photogate] from the list.

(2) Turn on the interface and run Capstone software.

The Photogate’s icon will be added to the panel and [Timer Setup] icon will appear in the [Tools] palette.

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Lab Manual

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(3-2) Create a timer.

Free Fall & Projectile Motion

Ver.20180302

β‘£ Make sure [Position] is checked.

Click [Timer Setup] in the [Tools] palette and follow the steps below.

β‘  Select [Pre-Configured Timer] and click [Next].

β‘€ Enter a suitable value in the [Flag Spacing].

β‘‘ Check [Photogate, Ch1] and click [Next].

You have two kinds of Picket Fences as follows.

PF#1 PF#2

Spacing

Description

0.05m

Opaque 0.02m + Transp. 0.03m

0.04m

Opaque 0.02m + Transp. 0.02m

β‘’ Select [Picket Fence].

β‘₯ Enter any name and finish the timer setup.

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(3-3) Create a graph display and a data table.

Free Fall & Projectile Motion

Ver.20180302

Click and drag the [Table] icon from the [Displays] palette into the workbook page. Select [Time(s)] for the first column

Click and drag the [Graph] icon from the [Displays] palette

and [Position(m)] for the second column.

into the workbook page.

A graph display will appear.

You now have two displays in the workbook page.

(4) Begin recording data.

Click [Record] in the [Controls] palette. Capstone begins You can select the measurement of each axis by clicking

recording all available data.

.

(4) Create and configure a timer. (6) Load the launcher. Click [Timer Setup] in the [Tools] palette and follow the steps below to create a timer.

Place the GREEN ball in the muzzle of the launcher. And

β‘’ Select [One Photogate (Single Flag)] for the type of timer.

then push the ball down the barrel with the ramrod until the

β‘£ Make sure [Speed] is checked.

trigger catches the MEDIUM RANGE setting of the piston.

β‘€ Enter the measured diameter of the ball for [Flag Width].

(The trigger will click into place.) You can use a different range setting, if required. If you cannot coke the piston due to a structural problem, pull and return the trigger while you are pushing the ramrod.

NOTE The Launcher has three range settings. The reference lines of the ramrod show the positions of each setting.

Remember, if you cock the piston with a ball in the piston, the piston is in the MEDIUM RANGE position when the first (left) line (not the middle line) of the ramrod reaches near the entrance of the muzzle,

CAUTION NEVER LOOK INTO THE MUZZLE of the launcher when it is loaded. Accidental shooting can cause blindness or serious loss of vision.

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Lab Manual

General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Free Fall & Projectile Motion

Ver.20180302

(7) Measure the initial speed of the ball.

(9) Place white paper and carbon paper on the box.

β‘  Click [Record] in the [Controls] palette.

β‘  Fire a test shot to locate where the ball hits.

β‘‘ Shoot the ball by pulling straight up the trigger.

β‘‘ Put a piece of white paper on the box at this location. β‘’ Put a piece of carbon paper (carbon-side down) on top of the white paper.

When the ball hits the carbon paper, it will leave a mark on the white paper underneath. β‘’ Click [Stop] and record the speed of the ball. β‘£ Repeat more than 3 times with the same range setting. β‘€ Find the average speed of the ball. The average represents the initial speed 𝑣𝑣0 of the projectile. 1st 𝑣𝑣result 𝑣𝑣AVG

2nd

3rd

…

𝑣𝑣0 = ___________(m/s)

(10) Begin experiment.

β‘  Fire three shots with the same range setting of step (7). (8) Adjust the height and the angle of the Launcher.

β‘‘ Carefully remove the carbon paper. β‘’ Use a measuring tape to measure the horizontal distance

The height and angle of inclination above the horizontal is adjusted by loosening the two thumbscrews and rotating the Launcher barrel to the desired angle. Use the plumb bob and

𝑅𝑅 from the muzzle to the dots.

β‘£ Repeat measuring 𝑅𝑅 for angles below.

the protractor on the label to select the angle. Tighten both

𝛼𝛼0

thumbscrews when the angle is set.

1st

2nd

25Β°

𝑅𝑅

3rd

AVG

35Β°

45Β°

55Β° 65Β°

β‘€ Calculate the error between the theoretical distance and the actual average distance for all angles.

𝑅𝑅 = (𝑣𝑣0 cos 𝛼𝛼0 )𝑑𝑑1 = Q

2𝑣𝑣02 sin 𝛼𝛼0 cos 𝛼𝛼0 𝑣𝑣02 = sin 2𝛼𝛼0 𝘨𝘨 𝘨𝘨

(16)

What angle give the maximum range of 𝑅𝑅?

What pairs of angles have a common range?

A

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Lab Manual

General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Free Fall & Projectile Motion

Ver.20180302

Appendix

1. Method of Least Squares Whenever we perform an experiment, we need to extract

The sum of all of the squares of the deviations is called the residual πœ’πœ’ 2 , given by

useful information from the collected data. We usually meas-

πœ’πœ’ 2 = ��𝑦𝑦𝑖𝑖 βˆ’ 𝑓𝑓(π‘₯π‘₯𝑖𝑖 )οΏ½

ure one variable under a variety of conditions with regard to a second variable. The method of least squares is a useful statistical technique to estimate a mathematical expression for the relationship between the two variables. Through series of observation, we get a series of 𝑛𝑛 meas-

urements of the pair (π‘₯π‘₯𝑖𝑖 , 𝑦𝑦𝑖𝑖 ), where 𝑖𝑖 is an index that runs from 1 to 𝑛𝑛. Suppose a certain mathematical model 𝑦𝑦 = 𝑓𝑓(π‘₯π‘₯)

best describes the relationship between π‘₯π‘₯𝑖𝑖 and 𝑦𝑦𝑖𝑖 . Here, we

have two value sets; 𝑦𝑦𝑖𝑖 is experimental value obtained

through series of observation, and 𝑓𝑓(π‘₯π‘₯𝑖𝑖 ) is the function value calculated by the model 𝑦𝑦 = 𝑓𝑓(π‘₯π‘₯).

Consider the distances (or deviations) between 𝑦𝑦𝑖𝑖 and

𝑓𝑓(π‘₯π‘₯𝑖𝑖 ) as shown below. If those deviations are as small as

possible, we can say the model 𝑦𝑦 = 𝑓𝑓(π‘₯π‘₯) is a really good model for the data. The method of least squares attempts to minimize the square of the deviations.

2

(1)

Suppose the collected data have a linear relationship, then the model 𝑓𝑓(π‘₯π‘₯) can be expressed in the general form 𝑓𝑓(π‘₯π‘₯) = π‘Žπ‘Ž + 𝑏𝑏𝑏𝑏

(2)

Substituting Eq. (2) into Eq, (1) yields

2

πœ’πœ’ 2 = ��𝑦𝑦𝑖𝑖 βˆ’ (π‘Žπ‘Ž + 𝑏𝑏π‘₯π‘₯𝑖𝑖 )οΏ½ = οΏ½(𝑦𝑦𝑖𝑖 βˆ’ π‘Žπ‘Ž βˆ’ 𝑏𝑏π‘₯π‘₯𝑖𝑖 )2

(3)

To find 𝑓𝑓(π‘₯π‘₯) that best fits the data, the residual should be

as small as possible, i.e. parameters π‘Žπ‘Ž and 𝑏𝑏 should be chosen so that they minimize πœ’πœ’ 2 .

Differentiating equation (3) with respect to π‘Žπ‘Ž and 𝑏𝑏 and

setting these differentials equal to zero produces the following equations for the optimum values of the parameters. πœ•πœ•πœ’πœ’ 2 = βˆ’2 οΏ½ 𝑦𝑦𝑖𝑖 + 2𝑏𝑏 οΏ½ π‘₯π‘₯𝑖𝑖 + 2π‘Žπ‘Žπ‘Žπ‘Ž = 0 πœ•πœ•πœ•πœ•

πœ•πœ•πœ’πœ’ 2 = βˆ’2 οΏ½ π‘₯π‘₯𝑖𝑖 𝑦𝑦𝑖𝑖 + 2π‘Žπ‘Ž οΏ½ π‘₯π‘₯𝑖𝑖 + 2𝑏𝑏 οΏ½ π‘₯π‘₯𝑖𝑖2 = 0 πœ•πœ•πœ•πœ•

(4)

Finally we obtain

π‘Žπ‘Ž =

οΏ½βˆ‘ π‘₯π‘₯𝑖𝑖2 οΏ½(βˆ‘ 𝑦𝑦𝑖𝑖 ) βˆ’ (βˆ‘ π‘₯π‘₯𝑖𝑖 )(βˆ‘ π‘₯π‘₯𝑖𝑖 𝑦𝑦𝑖𝑖 ) π‘›π‘›οΏ½βˆ‘ π‘₯π‘₯𝑖𝑖2 οΏ½ βˆ’ (βˆ‘ π‘₯π‘₯𝑖𝑖 )2

(5)

𝑛𝑛(βˆ‘ π‘₯π‘₯𝑖𝑖 𝑦𝑦𝑖𝑖 ) βˆ’ (βˆ‘ π‘₯π‘₯𝑖𝑖 )(βˆ‘ 𝑦𝑦𝑖𝑖 ) 𝑏𝑏 = π‘›π‘›οΏ½βˆ‘ π‘₯π‘₯𝑖𝑖2 οΏ½ βˆ’ (βˆ‘ π‘₯π‘₯𝑖𝑖 )2

Lab Office (Int’l Campus)

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General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Lab Manual

Free Fall & Projectile Motion

Ver.20180302

Result & Discussion

Your TA will inform you of the guidelines for writing the laboratory report during the lecture.

End of LAB Checklist

Please put your equipment in order as shown below.

β–‘ Delete your data files from your lab computer. β–‘ Turn off the Computer and the Interface. β–‘ Clamp the Projectile Launcher to the left end of the table. β–‘ Keep the Photogate Bracket assembled to the Photogate. β–‘ Keep the Carbon Paper in the plastic bag.

Lab Office (Int’l Campus)

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