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₯-π‘π‘ graph of the carβs position as a function

of time. The average velocity of the car equals the slope of the line ππ1 ππ2. But the average velocity during a time interval

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

canβt 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₯-π‘π‘ graph

(Fig. 2), the instantaneous velocity at any point is equal to the When a car moves from ππ1 to ππ2 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₯ -

π£π£π₯π₯ = lim

component of displacement βπ₯π₯ = π₯π₯2 β π₯π₯1 divided by the time

βπ‘π‘β0

interval βπ‘π‘ = π‘π‘2 β π‘π‘1 during which the displacement occurs. π£π£av-π₯π₯ =

Fig. 1

π₯π₯2 β π₯π₯1 βπ₯π₯ = π‘π‘2 β π‘π‘1 βπ‘π‘

Positions of a car at two times during its run.

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βπ₯π₯ ππππ = βπ‘π‘ ππππ

(2)

(1)

Fig. 2

The position of a car as a function of time.

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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 π‘π‘1 the object is at point ππ1 and

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

at point ππ2 and has velocity π£π£2π₯π₯ . So the velocity changes by

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

has π₯π₯-component of velocity π£π£1π₯π₯ , and at a later time π‘π‘2 it is

amount βπ£π£π₯π₯ = π£π£2π₯π₯ β π£π£1π₯π₯ during βπ‘π‘ = π‘π‘2 β π‘π‘1 . We define the

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

average acceleration of the object equals βπ£π£π₯π₯ divided by

βπ‘π‘.

ππav-π₯π₯ =

π£π£2π₯π₯ β π£π£1π₯π₯ βπ£π£π₯π₯ = π‘π‘2 β π‘π‘1 βπ‘π‘

(3)

We can now define instantaneous acceleration following

π£π£av-π₯π₯ =

π₯π₯ β π₯π₯0 π‘π‘

(6)

We can also get a second expression for π£π£avΛΜ΅π₯π₯ . In this case

the average π₯π₯-velocity for the time interval from 0 to π‘π‘ is simply the average of π£π£0π₯π₯ and π£π£π₯π₯ .

the same procedure that we used to define instantaneous

π£π£avΛΜ΅π₯π₯ =

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

π£π£0π₯π₯ + π£π£π₯π₯ 2

(7)

Substituting Eq. (5) into Eq. (7) yields πππ₯π₯ = lim

βπ‘π‘β0

βπ£π£π₯π₯ πππ£π£π₯π₯ = βπ‘π‘ ππππ

(4)

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

1 1 π£π£avΛΜ΅π₯π₯ = (π£π£0π₯π₯ + π£π£0π₯π₯ + πππ₯π₯ π‘π‘) = π£π£0π₯π₯ + πππ₯π₯ π‘π‘ 2 2

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

tion with constant acceleration. We can find the velocity π£π£π₯π₯ of

1 π₯π₯ = π₯π₯0 + π£π£0π₯π₯ π‘π‘ + πππ₯π₯ π‘π‘ 2 2

that motion using Eq. (3). We use π£π£0π₯π₯ for the π₯π₯-velocity at

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

becomes

π£π£π₯π₯ β π£π£0π₯π₯ πππ₯π₯ = π‘π‘ β 0

π£π£π₯π₯ = π£π£0π₯π₯ + πππ₯π₯ π‘π‘

or

(8)

(9)

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

(5)

is a constant π₯π₯-acceleration, the additional (1β2)πππ₯π₯ π‘π‘ 2 term

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

ing a constant π₯π₯-acceleration gives a slope to the π£π£π₯π₯ -π‘π‘ graph

(Fig. 4(b)).

Fig. 3

A π£π£π₯π₯ -π‘π‘ 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 bodyβs (a) π₯π₯-π‘π‘ graph and (b) π£π£π₯π₯ -π‘π‘ graph.

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The most familiar example of motion with constant acceleration is a body falling under the influence of the earthβs 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 πππ₯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-

πππ¦π¦ = βπ¨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 π‘π‘ = 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 π¨π¨. The approximate

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

its velocity components have the initial values π£π£0π₯π₯ = π£π£0 cos πΌπΌ0 and π£π£0π¦π¦ = π£π£0 sin πΌπΌ0 . From Eqs. (5), (9) and (10), we find π£π£π₯π₯ = π£π£0π₯π₯ + πππ₯π₯ π‘π‘ = π£π£0 cos πΌπΌ0

π£π£π¦π¦ = π£π£0π¦π¦ + πππ¦π¦ π‘π‘ = π£π£0 sin πΌπΌ0 β π¨π¨π¨π¨

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 π¦π¦-axis vertically upward. The π₯π₯-component of accelera-

tion is zero, and the π¦π¦-component is constant and equal to

βπ¨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 π¦π¦ = π¦π¦0 + π£π£0π¦π¦ π‘π‘ + πππ¦π¦ π‘π‘ 2 = (π£π£0 sin πΌπΌ0 )π‘π‘ β 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 + π£π£0π₯π₯ π‘π‘ + πππ₯π₯ π‘π‘ 2 = (π£π£0 cos πΌπΌ0 )π‘π‘ 2

value of π¨π¨ near the earthβs surface is 9.8 mβs2 .

given an initial velocity and then follows a path determined

(11)

(14)

The time π‘π‘1 when the projectile hits the ground is 1 0 = (π£π£0 sin πΌπΌ0 )π‘π‘1 β π¨π¨π‘π‘12 2

or

π‘π‘1 =

2π£π£0 sin πΌπΌ0 π¨π¨

(15)

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

stituting equation (15) into equation (13) yields

π π = (π£π£0 cos πΌπΌ0 )π‘π‘1 =

2π£π£02 sin πΌπΌ0 cos πΌπΌ0 π£π£02 = sin 2πΌπΌ0 π¨π¨ π¨π¨

(16)

In Eq. (16), the maximum value of sin 2πΌπΌ0 is 1. This occurs

when 2πΌπΌ0 = 90Β° or πΌπΌ0 = 45Β°. This angle gives the maximum

π π 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 π¦π¦-position, π¦π¦-velocity, and π¦π¦-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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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 (Intβl Campus)

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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 β0β and the LED lamp on the photogate goes on. When the beam is not blocked, the output signal

(5) Vernier Caliper

becomes β1β 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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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 Photogateβs icon will be added to the panel and [Timer Setup] icon will appear in the [Tools] palette.

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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 π£π£0 of the projectile. 1st π£π£result π£π£AVG

2nd

3rd

β¦

π£π£0 = ___________(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

π π from the muzzle to the dots.

β£ Repeat measuring π π for angles below.

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

πΌπΌ0

thumbscrews when the angle is set.

1st

2nd

25Β°

π π

3rd

AVG

35Β°

45Β°

55Β° 65Β°

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

π π = (π£π£0 cos πΌπΌ0 )π‘π‘1 = Q

2π£π£02 sin πΌπΌ0 cos πΌπΌ0 π£π£02 = sin 2πΌπΌ0 π¨π¨ π¨π¨

(16)

What angle give the maximum range of π π ?

What pairs of angles have a common range?

A

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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 ππ 2 , given by

useful information from the collected data. We usually meas-

ππ 2 = οΏ½οΏ½π¦π¦ππ β ππ(π₯π₯ππ )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 ππ meas-

urements of the pair (π₯π₯ππ , π¦π¦ππ ), where ππ is an index that runs from 1 to ππ. Suppose a certain mathematical model π¦π¦ = ππ(π₯π₯)

best describes the relationship between π₯π₯ππ and π¦π¦ππ . Here, we

have two value sets; π¦π¦ππ is experimental value obtained

through series of observation, and ππ(π₯π₯ππ ) is the function value calculated by the model π¦π¦ = ππ(π₯π₯).

Consider the distances (or deviations) between π¦π¦ππ and

ππ(π₯π₯ππ ) as shown below. If those deviations are as small as

possible, we can say the model π¦π¦ = ππ(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 ππ(π₯π₯) can be expressed in the general form ππ(π₯π₯) = ππ + ππππ

(2)

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

2

ππ 2 = οΏ½οΏ½π¦π¦ππ β (ππ + πππ₯π₯ππ )οΏ½ = οΏ½(π¦π¦ππ β ππ β πππ₯π₯ππ )2

(3)

To find ππ(π₯π₯) that best fits the data, the residual should be

as small as possible, i.e. parameters ππ and ππ should be chosen so that they minimize ππ 2 .

Differentiating equation (3) with respect to ππ and ππ and

setting these differentials equal to zero produces the following equations for the optimum values of the parameters. ππππ 2 = β2 οΏ½ π¦π¦ππ + 2ππ οΏ½ π₯π₯ππ + 2ππππ = 0 ππππ

ππππ 2 = β2 οΏ½ π₯π₯ππ π¦π¦ππ + 2ππ οΏ½ π₯π₯ππ + 2ππ οΏ½ π₯π₯ππ2 = 0 ππππ

(4)

Finally we obtain

ππ =

οΏ½β π₯π₯ππ2 οΏ½(β π¦π¦ππ ) β (β π₯π₯ππ )(β π₯π₯ππ π¦π¦ππ ) πποΏ½β π₯π₯ππ2 οΏ½ β (β π₯π₯ππ )2

(5)

ππ(β π₯π₯ππ π¦π¦ππ ) β (β π₯π₯ππ )(β π¦π¦ππ ) ππ = πποΏ½β π₯π₯ππ2 οΏ½ β (β π₯π₯ππ )2

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Lab Manual

Free Fall & Projectile Motion

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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.

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