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

Lab Manual

Free Fall & Projectile Motion Ver.20180302

Lab Office (Int’l Campus)

Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) / 15

[International Campus]

Free Fall and Projectile Motion

Investigate the motions of a freely falling body and a projectile under the influence of gravity.

Find the acceleration due to gravity.

When a car moves from 𝑃𝑃1 to 𝑃𝑃2 in the +𝑥𝑥-direction as in

figure 1, the 𝑥𝑥 -component of average velocity is the 𝑥𝑥 -

component of displacement ∆𝑥𝑥 = 𝑥𝑥2 − 𝑥𝑥1 divided by the time

interval ∆𝑡𝑡 = 𝑡𝑡2 − 𝑡𝑡1 during which the displacement occurs.

𝑣𝑣av-𝑥𝑥 =𝑥𝑥2 − 𝑥𝑥1𝑡𝑡2 − 𝑡𝑡1

=∆𝑥𝑥∆𝑡𝑡 (1)

Fig. 1 Positions of a car at two times during its run.

Figure 2 is the 𝑥𝑥-𝑡𝑡 graph of the car’s position as a function

of time. The average velocity of the car equals the slope of

the line 𝑝𝑝1𝑝𝑝2. But the average velocity during a time interval

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

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

cific instant of time or specific point along the path.

The instantaneous velocity is the limit of the average veloci-

ty as the time interval approaches zero. On the 𝑥𝑥-𝑡𝑡 graph

(Fig. 2), the instantaneous velocity at any point is equal to the

slope of the tangent to the curve at that point.

𝑣𝑣𝑥𝑥 = lim∆𝑡𝑡→0

∆𝑥𝑥∆𝑡𝑡 =

𝑑𝑑𝑥𝑥𝑑𝑑𝑡𝑡 (2)

Fig. 2 The position of a car as a function of time.

Objective

Theory

----------------------------- Reference --------------------------

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

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

2.2 Instantaneous Velocity (p.61~64)

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

2.4 Motion with Constant Acceleration (p.69~74)

2.5 Freely Falling Bodies (p.74~77)

3.3 Projectile Motion (p.99~106)

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


Lab Manual




Acceleration describes the rate of change of velocity with

time. Suppose that at time 𝑡𝑡1 the object is at point 𝑃𝑃1 and

has 𝑥𝑥-component of velocity 𝑣𝑣1𝑥𝑥, and at a later time 𝑡𝑡2 it is

at point 𝑃𝑃2 and has velocity 𝑣𝑣2𝑥𝑥. So the velocity changes by

amount ∆𝑣𝑣𝑥𝑥 = 𝑣𝑣2𝑥𝑥 − 𝑣𝑣1𝑥𝑥 during ∆𝑡𝑡 = 𝑡𝑡2 − 𝑡𝑡1. We define the

average acceleration of the object equals ∆𝑣𝑣𝑥𝑥 divided by

∆𝑡𝑡.

𝑎𝑎av-𝑥𝑥 =𝑣𝑣2𝑥𝑥 − 𝑣𝑣1𝑥𝑥𝑡𝑡2 − 𝑡𝑡1

=∆𝑣𝑣𝑥𝑥∆𝑡𝑡 (3)

We can now define instantaneous acceleration following

the same procedure that we used to define instantaneous

velocity. The instantaneous acceleration is the limit of the

average acceleration as the time interval approaches zero.

𝑎𝑎𝑥𝑥 = lim∆𝑡𝑡→0

∆𝑣𝑣𝑥𝑥∆𝑡𝑡 =

𝑑𝑑𝑣𝑣𝑥𝑥𝑑𝑑𝑡𝑡 (4)

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

tion with constant acceleration. We can find the velocity 𝑣𝑣𝑥𝑥 of

that motion using Eq. (3). We use 𝑣𝑣0𝑥𝑥 for the 𝑥𝑥-velocity at

𝑡𝑡 = 0; the 𝑥𝑥-velocity at the later time 𝑡𝑡 is 𝑣𝑣𝑥𝑥. Then Eq. (3)

becomes

𝑎𝑎𝑥𝑥 =𝑣𝑣𝑥𝑥 − 𝑣𝑣0𝑥𝑥𝑡𝑡 − 0 or

𝑣𝑣𝑥𝑥 = 𝑣𝑣0𝑥𝑥 + 𝑎𝑎𝑥𝑥𝑡𝑡 (5)

Fig. 3 A 𝑣𝑣𝑥𝑥-𝑡𝑡 graph for an object moving with constant

acceleration on 𝑥𝑥-axis.

We can also derive an equation for the position 𝑥𝑥 as a func-

tion of time using Eqs. (1) and (5) when the 𝑥𝑥-acceleration is

constant. With the initial position 𝑥𝑥0 at time 𝑡𝑡 = 0 and the

position 𝑥𝑥 at the later time 𝑡𝑡, Eq. (1) becomes

𝑣𝑣av-𝑥𝑥 =𝑥𝑥 − 𝑥𝑥0𝑡𝑡 (6)

We can also get a second expression for 𝑣𝑣av˗�̵�𝑥. In this case

the average 𝑥𝑥-velocity for the time interval from 0 to 𝑡𝑡 is

simply the average of 𝑣𝑣0𝑥𝑥 and 𝑣𝑣𝑥𝑥.

𝑣𝑣av˗̵𝑥𝑥 =𝑣𝑣0𝑥𝑥 + 𝑣𝑣𝑥𝑥

2 (7)

Substituting Eq. (5) into Eq. (7) yields

𝑣𝑣av˗̵𝑥𝑥 =12

(𝑣𝑣0𝑥𝑥 + 𝑣𝑣0𝑥𝑥 + 𝑎𝑎𝑥𝑥𝑡𝑡) = 𝑣𝑣0𝑥𝑥 +12𝑎𝑎𝑥𝑥𝑡𝑡

(8)

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

𝑥𝑥 = 𝑥𝑥0 + 𝑣𝑣0𝑥𝑥𝑡𝑡 +12𝑎𝑎𝑥𝑥𝑡𝑡

2 (9)

Figure 4 shows the graphs of Eq. (9) and Eq. (5). If there is

zero 𝑥𝑥-acceleration, the 𝑥𝑥-𝑡𝑡 graph is a straight line; if there

is a constant 𝑥𝑥-acceleration, the additional (1 2⁄ )𝑎𝑎𝑥𝑥𝑡𝑡2 term

curves the graph into a parabola (Fig. 4(a)). Also, if there is

zero 𝑥𝑥-acceleration, the 𝑣𝑣𝑥𝑥-𝑡𝑡 graph is a horizontal line; add-

ing a constant 𝑥𝑥-acceleration gives a slope to the 𝑣𝑣𝑥𝑥-𝑡𝑡 graph

(Fig. 4(b)).

Fig. 4 How a constant 𝑥𝑥-acceleration affects a body’s

(a) 𝑥𝑥-𝑡𝑡 graph and (b) 𝑣𝑣𝑥𝑥-𝑡𝑡 graph.


Lab Manual




The most familiar example of motion with constant accelera-

tion is a body falling under the influence of the earth’s gravita-

tional attraction. If the distance of the fall is small compared

with the radius of the earth, and if the effects of the air can be

neglected, all bodies fall with the same downward accelera-

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

of falling bodies separated by equal time intervals. The red

ball is dropped from rest. There are equal time intervals be-

tween 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 con-

stant. This acceleration is called the acceleration due to

gravity. We denote its magnitude with 𝘨𝘨. The approximate

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

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

given an initial velocity and then follows a path determined

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

glect the effects of air resistance.) The motion of the yellow

ball in Fig. 5 is two-dimensional. We will call the plane of mo-

tion the 𝑥𝑥𝑥𝑥-coordinate plane, with the 𝑥𝑥-axis horizontal and

the 𝑥𝑥-axis vertically upward. The 𝑥𝑥-component of accelera-

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

−𝘨𝘨. So we can analyze projectile motion as a combination of

horizontal motion with constant velocity and vertical motion

with constant acceleration.

Fig. 5 The red ball is dropped from rest, and the yellow ball is

simultaneously projected horizontally; successive im-ages in this stroboscopic photograph are separated by equal time intervals. At any given time, both balls have the same 𝑥𝑥-position, 𝑥𝑥-velocity, and 𝑥𝑥-acceleration, despite having different 𝑥𝑥-positions and 𝑥𝑥-velocities.

We can then express all the vector relationships for the posi-

tion, velocity, and acceleration by separate equations

𝑎𝑎𝑥𝑥 = 0 𝑎𝑎𝑦𝑦 = −𝘨𝘨 (10)

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

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

projectile is at the point (𝑥𝑥0,𝑥𝑥0) = (0, 0) and that at this time

its velocity components have the initial values 𝑣𝑣0𝑥𝑥 = 𝑣𝑣0 cos𝛼𝛼0

and 𝑣𝑣0𝑦𝑦 = 𝑣𝑣0 sin𝛼𝛼0. From Eqs. (5), (9) and (10), we find

𝑣𝑣𝑥𝑥 = 𝑣𝑣0𝑥𝑥 + 𝑎𝑎𝑥𝑥𝑡𝑡 = 𝑣𝑣0 cos𝛼𝛼0 (11)

𝑣𝑣𝑦𝑦 = 𝑣𝑣0𝑦𝑦 + 𝑎𝑎𝑦𝑦𝑡𝑡 = 𝑣𝑣0 sin𝛼𝛼0 − 𝘨𝘨𝑡𝑡 (12)

𝑥𝑥 = 𝑥𝑥0 + 𝑣𝑣0𝑥𝑥𝑡𝑡 +12𝑎𝑎𝑥𝑥𝑡𝑡

2 = (𝑣𝑣0 cos𝛼𝛼0)𝑡𝑡 (13)

𝑥𝑥 = 𝑥𝑥0 + 𝑣𝑣0𝑦𝑦𝑡𝑡 +12𝑎𝑎𝑦𝑦𝑡𝑡

2 = (𝑣𝑣0 sin𝛼𝛼0)𝑡𝑡 −12𝘨𝘨𝑡𝑡

2 (14)

The time 𝑡𝑡1 when the projectile hits the ground is

0 = (𝑣𝑣0 sin𝛼𝛼0)𝑡𝑡1 −12 𝘨𝘨𝑡𝑡1

2 or 𝑡𝑡1 =2𝑣𝑣0 sin𝛼𝛼0

𝘨𝘨 (15)

The horizontal range 𝑅𝑅 is the value of 𝑥𝑥 at this time. Sub-

stituting equation (15) into equation (13) yields

𝑅𝑅 = (𝑣𝑣0 cos𝛼𝛼0)𝑡𝑡1 =2𝑣𝑣02 sin𝛼𝛼0 cos𝛼𝛼0

𝘨𝘨 =𝑣𝑣02

𝘨𝘨 sin 2𝛼𝛼0 (16)

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

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

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

Fig. 6 The trajectory of a projectile is a combination of horizontal motion with constant velocity and vertical motion with constant acceleration.


Lab Manual




1. List

Item(s) Qty. Description

PC / Software Data Analysis: Capstone

1 Records, displays and analyzes data measured by var-ious sensors.

Interface

1 Data acquisition interface designed for use with various 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 Multi-clamp

1 1

Provide stable support for experiment set-ups.

Support Rod (600mm)

1 Provides stable support for experiment set-ups.

Cushioned Baskets

1 Absorb shock on impact.

Picket Fence set

1 set

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 Launches a ball at any angle from zero to ninety de-grees with three range settings.

Photogate Bracket

1 Mounts the Photogate on the Projectile Launcher.

Equipment


Lab Manual




Item(s) Qty. Description

Projectile (Green Plastic Ball)

1 Green plastic ball which can be loaded into the Projec-tile Launcher.

Table Clamp

1 Clamps the Projectile Launcher to a lab table.

Carbon Paper White Paper

1 1

Leaves a mark when a projectile ball hits it. (White Paper is not provided.)

Box

1 Provides a horizontal surface so the projectile ball can reaches the same level as the muzzle of the Projectile Launcher.

Measuring Tape

1 Measures distance.

Vernier Caliper

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

2. Details

(1) Interface

The 850 Universal Interface is a data acquisition interface

designed for use with various sensors to measure physical

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.

(2) Capstone: Data Acquisition and Analysis Software

The Capstone software records, displays and analyzes data

measured by the sensor connected to the 850 interface. It

also controls the built-in signal generator of the interface.


Lab Manual




(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-

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

face.

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

photogate becomes ‘0’ and the LED lamp on the photogate

goes on. When the beam is not blocked, the output signal

becomes ‘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 pro-

tractor and plumb-bob give and accurate way to measure the

angle of inclination.

The Projectile Launcher has three range settings so that

balls can be launched with three different initial speeds. One

or two Photogates can be attached to the Projectile Launcher

using the Photogate Bracket so that the photogates can

measure the initial speed of the ball.

(5) Vernier Caliper

① 22 mm is to the immediate left of the zero on the vernier

scale. Hence, the main scale reading is 22 mm.

② Look closely for and alignment of the scale lines of the

main scale and vernier scale. In the figure, the aligned (13th)

line corresponds to 0.65 mm (= 0.05 × 13).

③ The final measurement is given by the sum of the two

readings. This gives 22.65 mm (= 22 + 0.65).


Lab Manual




Experiment 1. Free Fall

(1) Set up equipment as below.

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

The interface is automatically detected by Capstone. Click

[Hardware Setup] in the [Tools] palette to configure the inter-

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.

The Photogate’s icon will be added to the panel and [Timer

Setup] icon will appear in the [Tools] palette.

Procedure


Lab Manual




(3-2) Create a timer.

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

below.

① Select [Pre-Configured Timer] and click [Next].

② Check [Photogate, Ch1] and click [Next].

③ Select [Picket Fence].

④ Make sure [Position] is checked.

⑤ Enter a suitable value in the [Flag Spacing].

You have two kinds of Picket Fences as follows.

Spacing Description

PF#1 0.05m Opaque 0.02m + Transp. 0.03m

PF#2 0.04m Opaque 0.02m + Transp. 0.02m

⑥ Enter any name and finish the timer setup.


Lab Manual




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

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

into the workbook page.

A graph display will appear.

You can select the measurement of each axis by clicking

<Select Measurement>. Select [Time(s)] for the 𝑥𝑥-axis and

[Position(m)] of the Picket Fence for the 𝑥𝑥-axis.

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

into the workbook page. Select [Time(s)] for the first column

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

You now have two displays in the workbook page.

(4) Begin recording data.

Click [Record] in the [Controls] palette. Capstone begins

recording all available data.

[Record] button will toggle to [Stop]. You can stop data col-

lection by clicking [Stop].

Collected data are stored in memory and appear in all dis-

plays. The data run is listed in the legend for each display.

(You can delete any run by clicking [Delete Last Run] or drop-

down menu in the [Controls] panel.)


Lab Manual




(5) Drop a Picket Fence.

Make sure the opaque bands of the Picket Fence block the

infrared beam of the Photogate during they pass through the

Photogate.

(6) Stop data collection.

Click the [Stop] button to stop data collection.

(7) Analyze the data.

Time(s) Position(m)

1 𝑡𝑡1 𝑥𝑥1



… … …

Calculate the average speed at each interval, and plot 𝑣𝑣-𝑡𝑡

graph.

𝑡𝑡 𝑣𝑣

1~2 𝑡𝑡1 + 𝑡𝑡2

2

𝑥𝑥2 − 𝑥𝑥1𝑡𝑡2 − 𝑡𝑡1

2~3 𝑡𝑡2 + 𝑡𝑡3

2


3~4 𝑡𝑡3 + 𝑡𝑡4

2


… … …

Find the slope of 𝑣𝑣-𝑡𝑡 graph using the method of least

squares (refer to the appendix). The acceleration of the Pick-

et Fence is equal to the slope of the 𝑣𝑣-𝑡𝑡 graph.

(8) Repeat experiments.

① Repeat measurement with the same Picket Fence (more

than three times).

1st 2nd 3rd …

𝑎𝑎result

𝑎𝑎AVG

𝘨𝘨 9.8 m/s

② Repeat measurement using the other Picket Fence.

Change [Flag Spacing] parameter. See step (3-2)-⑤.

Save the data file with a different name for each Picket

Fence. If you change [Flag Spacing], all premeasured data

will be recalculated.


Lab Manual




Experiment 2. Projectile Motion

(1) Set up equipment.

Attach the Photogate Bracket to the Launcher and attach

the Photogate to the Bracket.

Align the square nuts of the Bracket with the T-shaped slot

on the bottom of the Launcher barrel and slide the nuts into

the slot until the Photogate is as close to the muzzle as pos-

sible without blocking the Photogate beam.

When the ball passes the Photogate, the initial speed of the

ball is calculated as

Speed =Moving distance (= diameter)of the ball

Time interval the ball blocks the Photogate

Thus, make sure the Photogate beam is exactly on the path

of the center of the ball. Tighten the Bracket thumbscrew to

secure the Photogate in place. (The Photogate could be

slightly shifted on impact. Whenever you shoot the ball, make

sure the Photogate is in position.) Using the table clamps,

clamp the Launcher to the left end of the table. Connect the

Photogate to the interface.

(2) Measure the diameter of the ball.

CAUTION

NEVER LOOK INTO THE MUZZLE of the Projectile

Launcher when it is loaded. Accidental shooting could

cause blindness or serious loss of vision.


Lab Manual




(3) Add a Photogate.

Click the port which you plugged the Photogate into and se-

lect [Photogate] from the list.

(4) Create and configure a timer.

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

below to create a timer.

③ Select [One Photogate (Single Flag)] for the type of timer.

④ Make sure [Speed] is checked.

⑤ Enter the measured diameter of the ball for [Flag Width].

(5) Create and configure a digital meter.

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

into the workbook page. Select [Speed(m/s)] for <Select

Measurement>.

(6) Load the launcher.

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

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

trigger catches the MEDIUM RANGE setting of the piston.

(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 pis-

ton, 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 blind-

ness or serious loss of vision.


Lab Manual




(7) Measure the initial speed of the ball.

① Click [Record] in the [Controls] palette.

② Shoot the ball by pulling straight up the trigger.

③ Click [Stop] and record the speed of the ball.

④ Repeat more than 3 times with the same range setting.

⑤ Find the average speed of the ball. The average repre-

sents the initial speed 𝑣𝑣0 of the projectile.

1st 2nd 3rd …

𝑣𝑣result

𝑣𝑣AVG 𝑣𝑣0 = ___________(m/s)

(8) Adjust the height and the angle of the Launcher.

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

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

thumbscrews when the angle is set.

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

① Fire a test shot to locate where the ball hits.

② Put a piece of white paper on the box at this location.

③ 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.

(10) Begin experiment.

① Fire three shots with the same range setting of step (7).

② Carefully remove the carbon paper.

③ Use a measuring tape to measure the horizontal distance

𝑅𝑅 from the muzzle to the dots.

④ Repeat measuring 𝑅𝑅 for angles below.

𝛼𝛼0 𝑅𝑅

1st 2nd 3rd AVG

25°

35°

45°

55°

65°

⑤ Calculate the error between the theoretical distance and

the actual average distance for all angles.

𝑅𝑅 = (𝑣𝑣0 cos𝛼𝛼0)𝑡𝑡1 =2𝑣𝑣02 sin𝛼𝛼0 cos𝛼𝛼0

𝘨𝘨 =𝑣𝑣02

𝘨𝘨 sin 2𝛼𝛼0 (16)

Q What angle give the maximum range of 𝑅𝑅?

What pairs of angles have a common range?

A


Lab Manual




1. Method of Least Squares

Whenever we perform an experiment, we need to extract

useful information from the collected data. We usually meas-

ure one variable under a variety of conditions with regard to a

second variable. The method of least squares is a useful sta-

tistical 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 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖), where 𝑖𝑖 is an index that runs

from 1 to 𝑛𝑛. Suppose a certain mathematical model 𝑥𝑥 = 𝑓𝑓(𝑥𝑥)

best describes the relationship between 𝑥𝑥𝑖𝑖 and 𝑥𝑥𝑖𝑖. Here, we

have two value sets; 𝑥𝑥𝑖𝑖 is experimental value obtained

through series of observation, and 𝑓𝑓(𝑥𝑥𝑖𝑖) is the function value

calculated by the model 𝑥𝑥 = 𝑓𝑓(𝑥𝑥).

Consider the distances (or deviations) between 𝑥𝑥𝑖𝑖 and

𝑓𝑓(𝑥𝑥𝑖𝑖) as shown below. If those deviations are as small as

possible, we can say the model 𝑥𝑥 = 𝑓𝑓(𝑥𝑥) is a really good

model for the data. The method of least squares attempts to

minimize the square of the deviations.

The sum of all of the squares of the deviations is called the

residual 𝜒𝜒2, given by

𝜒𝜒2 = ��𝑥𝑥𝑖𝑖 − 𝑓𝑓(𝑥𝑥𝑖𝑖)�2 (1)

Suppose the collected data have a linear relationship, then

the model 𝑓𝑓(𝑥𝑥) can be expressed in the general form

𝑓𝑓(𝑥𝑥) = 𝑎𝑎 + 𝑏𝑏𝑥𝑥 (2)

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

𝜒𝜒2 = ��𝑥𝑥𝑖𝑖 − (𝑎𝑎 + 𝑏𝑏𝑥𝑥𝑖𝑖)�2 = �(𝑥𝑥𝑖𝑖 − 𝑎𝑎 − 𝑏𝑏𝑥𝑥𝑖𝑖)2 (3)

To find 𝑓𝑓(𝑥𝑥) that best fits the data, the residual should be

as small as possible, i.e. parameters 𝑎𝑎 and 𝑏𝑏 should be

chosen so that they minimize 𝜒𝜒2.

Differentiating equation (3) with respect to 𝑎𝑎 and 𝑏𝑏 and

setting these differentials equal to zero produces the following

equations for the optimum values of the parameters.

𝜕𝜕𝜒𝜒2

𝜕𝜕𝑎𝑎 = −2�𝑥𝑥𝑖𝑖 + 2𝑏𝑏�𝑥𝑥𝑖𝑖 + 2𝑎𝑎𝑎𝑎 = 0

𝜕𝜕𝜒𝜒2

𝜕𝜕𝑏𝑏 = −2�𝑥𝑥𝑖𝑖𝑥𝑥𝑖𝑖 + 2𝑎𝑎�𝑥𝑥𝑖𝑖 + 2𝑏𝑏�𝑥𝑥𝑖𝑖2 = 0

(4)

Finally we obtain

𝑎𝑎 =�∑𝑥𝑥𝑖𝑖2�(∑𝑥𝑥𝑖𝑖) − (∑𝑥𝑥𝑖𝑖)(∑𝑥𝑥𝑖𝑖𝑥𝑥𝑖𝑖)

𝑛𝑛�∑𝑥𝑥𝑖𝑖2� − (∑𝑥𝑥𝑖𝑖)2

𝑏𝑏 =𝑛𝑛(∑𝑥𝑥𝑖𝑖𝑥𝑥𝑖𝑖) − (∑𝑥𝑥𝑖𝑖)(∑𝑥𝑥𝑖𝑖)

𝑛𝑛�∑𝑥𝑥𝑖𝑖2� − (∑𝑥𝑥𝑖𝑖)2

(5)

Appendix


Lab Manual




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

Please put your equipment in order as shown below.

□ Delete your data files from your lab computer.

□ Turn off the Computer and the Interface.

□ Clamp the Projectile Launcher to the left end of the table.

□ Keep the Photogate Bracket assembled to the Photogate.

□ Keep the Carbon Paper in the plastic bag.

Result & Discussion

End of LAB Checklist

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