Physics Laboratory 1
Last modified : 2010.2.8
Experiment 2. A parabolic motion
Purpose of Experiment
Mechanics is the oldest field in physics where motion of a body is
studied. Kinetics describes motion, and dynamics connects the
relevant forces and the properties of a moving body with motion.
Getting the position of a body with respect to time, i.e. moving path
is basic, for example, a one dimensional motion in the “Newton’s
apple” experiment and a two dimensional motion like a parabolic
motion. in the “Newton’s apple” the direction in which the
gravitational force is acting and the direction of motion are the same,
thus we calculated the path in one coordinate only. However, since
a parabolic motion comes from the combination of a constant
velocity motion on the two dimensional plane (perpendicular plane)
and a constant acceleration motion in the perpendicular direction,
motion of a body is completely described when the positional
information of two coordinates as time flows. Galilei first discovered
that the projectile undergoes a parabolic motion.
Experiment Outline
- describe an ideal two dimensional motion ignoring friction and
rotational motion.
- investigate the case when a body is discharged at an angle and
the motion.
- see the effect of friction on motion using the ball with high air
friction.
- repeat the one dimensional free falling experiment like ‘Newton’s
apple’ and verify the reliability of measured data of the device.
Experiment method
The following devices are prepared for the experiment. (in the
parentheses are the numbers prepared)
A computer (1)
A CCD camera (1)
A reference ruler (1)
A firing device (1)
A plastic rod (1)
Beads
If you need anything else, ask the teaching assistant, visit the
experiment preparation room (19-111) or prepare them yourself.
There’s no specification of the experiment procedure although the
following is recommended.
Before the experiment, capture your own face using the
CCE and put it on the cover of your report.
1) see what trajectories a body discharged at various
angles takes and obtain the trace of the body as a function
of time.
① set up the apparatuses as in the movie. Set up the camera so that
the motion is observed from the side. It must be in the middle of
where the firing device is and where the bead would fall. Put the
reference ruler on the same plane (where motion occurs) of the
discharger. (think about why we put the camera and the ruler there
and discuss it in the result analysis.)
Turn on the computer and run “I-CA”. click [file-camera
configuration] on the menu and make sure the CCD screen is turned
on. (movie)
* Tip. Camera configuration
1. After running I-CA program, select the camera
configuration in the file menu.
2. Push the menu of the remote control till the setup
menu pops up.
3. Select the ALS/AES on the third row.
4. Select the LEVEL -FIX- OFF on the last row.
5. Select 1/250
6. Push BACK on the remote control and return to the
previous.
7. Select AGS/SENS
8. Select LIGHT and NORMAL
9. Select SENS
10. Select X32 (32 times)
11. Push BACK twice to exit
Refer to CCC screen adjustment(movie) and finish the initial device
configuration. Adjust the discharger at the known angle. The
discharger can be adjusted to three levels, but we use only first two
levels. Insert the reference ruler into the discharger for coordinate
configuration and click [file-capture screen] and record the movie
for a short time. (movie) Click [file-coordinates setup] and open the
movie just recorded and configure the coordinates.([starting point
setup]-[end point setup]-[input length]-[next]-[point of
reference(origin) setup]-[confirm])
② click [file-capture screen] and specify the data saving path and
shoot the bead to start the experiment. (movie) After saving data,
select [file-analyze] and analyze the saved data. ([select the first
frame]-[select the last frame]-[select the object]-[start analysis]-[save
the analyzed data]) (movie)
③ after the analysis, save the data and you can see the image file
used for analysis and the location data file of the object on the
screen. (x and y coordinates as time flows, and Vx and Vy for
each frame interval)
④ you can draw a graph obtained using Excel. (if you would like
open data using Origin, open the data using Excel and select the
file type as [text(delimited by tabs)] and save it and then open
it)
⑤ you can obtain a graph where the x coordinate of the bead is
linearly proportional to time and since the discharging angle in
known, you can calculate the initial velocity. The velocity of x
and y coordinates over time is the average speed between the
two measured frames. Are the initial velocities from (a) and (c)
the same? If not, why? The y coordinate of the bead is in the
shape of a second order curve over time and since the time
when y reaches the maximum can be calculated, gravitational
acceleration can be obtained using the initial velocity previously
calculated. Is it the same as the gravitational acceleration
generally known? How does Vx change over time? Compare the
result with the theoretically known value and discuss why the
result is so.
⑥
Verify that you get the same result (superimpose the graphs
with each other and do the fitting) by performing the experiment
at the same angle more than three times and obtain the average
and stddev. Repeat the experiment by varying the shooting angle
(more than five different angles) including 45o. Obtain the flying
time(T) before the projectile touches the ground, the horizontal
displacement(R) and the maximum displacement(Rmax) when the
shooting angle is 45 o from the graph and compare them with the
theoretical values. If they differ, discuss the reason.
2) comparison using the balls of different masses
Repeat the experiment using more than two of the beads with
different masses. Does the mass of bead affect the parabolic motion?
If so, discuss why. If not, what result do you get when the initial
velocities of all the beads are the same?
3) Comparison with the ‘Newton’s apple’ experiment
Perform a free falling experiment using more than two of the beads
of different masses and compare the result with that of the
‘Newton’s apple’. Is the gravitational acceleration same? Is the
mechanical energy conserved? Which case fits the theory better?
Why is that so? (discuss the possible difference between the
measurement method of this experiment and that of the ‘Newton’s
apple’ experiment.) obtain the coefficient of restitution between the
bead and the floor.
(4) collision between the bead with parabolic motion and the bead
with free falling motion
Point the shooter to the bead for free falling motion and set it in
falling motion at the same time with discharge. Observe the collision
between two balls. (do not use any program)
background theory
[fig. 1] each directional component of the initial velocity is
v0x=v0cosθ, v0y=v0sinθ
Let us describe the two dimensional motion of a body
discharged at some angle with the floor by solving of the
second order differential equation. As in the figure, the
force acting on a body discharged at the initial angle V0
with the floor at t=0+ is just the gravitational force in the
direction of –y. (the external shooting force acts on it only
until t=0)
That is, there’s no force in the direction of x and thus no change of
speed so acceleration is zero. However, there’s gravitational force
in the direction of –y. thus you can obtain the following using the
first law of Newton. (F=ma)
(1)
(2)
Integrating (1) and (2), you can obtain each x and y component of
the velocity over time.
(3)
(when t=0, the x component of the velocity is v0x=v0cosθ and you
can get the constant of integration C1) that is, the velocity is
constant since there’s no force in the direction of x. Integrating it
again, you can obtain the change of x coordinate over time.
(4)
(5)
When t=0, x(0)=x0 and you can get the constant of integration
C2=x0. The y coordinate over time can be obtained in the same way.
(6)
(7)
(8)
(9)
Rearrange the equation (5) with respect to time t, plug it in the
equation (8) and rearrange it then you can get the trajectory of the
body.
You can obtain the flying time(T) before the projectile touches the
ground by calculating the time when y(t)=0. (i.e. by solving a
quadratic equation)
(10)
(11)
The horizontal displacement(R) cane be calculated by plugging the
above (11) into the equation (5). (Do it yourself.) if you obtained the
answer, estimate the shooting angle when the horizontal
displacement is maximum(Rmax).
The maximum horizontal displacement is as follows
(12)
Now consider the friction effect of air. In general, a body with
speed v feels friction force as follows
(1<α<3, b is constant)
(13)
Suppose the friction force is proportional to speed then the
equations (1) and (2) become (k=b/m)
(14)
(15)
Integrating these, the velocity and the coordinates over time can be
obtained.
(16
)
(17)
(18)
After a long time, from the equation (16), the speed reaches a
certain terminal speed. (vy= -g/k) from this, the coefficient of
friction b is obtained.
references
※ Questions
① in the case of a rifle in a western film, as the barrel gets longer,
the time under the influence of force gets longer so the initial
ejecting speed of the bullet increases and it flies longer. When you
shoot the rifles with a longer barrel and a shorter barrel horizontally,
from which rifle does the bullet fly longer? Consider the case of the
experiment (4) together.
② theoretically, the maximum range is accomplished when the
shooting angle is 45. That is, at the angle larger or smaller than this,
the range becomes smaller than Rmax. thus there are two shooting
angles that can reach a distance R. Obtain two possible evaluation
angle of fire when the bead touches the ground at 50cm distance.
(Assume the height of the discharger’s entry as floor. Ignore air
friction as always.) [Use the initial speed obtained from the
experiment (1)] The sum of two angles is constant. Obtain the sum.

A method of processing the measurement data

A method of analysis using graphs

Isaac Newton – universal gravitation at the garden

Galileo galilei – father of experimental physics who paved the way for
Newton

The effect of friction due to the air on falling

Korea Research Institute of Standards and Science (KRISS)

National Institute of Standards and Technology (NIST)


Acceleration Due to Gravity


Acceleration Due to Gravity-Lecture Notes
※a site with various applets of projectile motion
http://physics.new21.org/pages/1/D/60/
Return to Index