MOTION WITH CONSTANT ACCELERATION
Alexander Sapozhnikov, Brooklyn College CUNY, New York, [email protected]
Objectives
• To study relation between position, velocity, and acceleration for motion with
constant acceleration.
• To calculate the acceleration for gravity.
• To study two-dimension motion.
Apparatus
An air table with blower, a puck, a meter stick, a rectangular wooden block, and a
computer with a webcam, a printer, an interface, and Logger Pro software.
Theory
In physics the motion of an object is defined as change of its position in time. The
motion is characterized by values of velocity and acceleration. Whereas the velocity is
the rate of change of the position, the acceleration is the rate of change of the velocity.
The velocity and acceleration can be defined
x − x1
v= 2
(1)
t 2 − t1
v −v
(2)
a= 2 1
t 2 − t1
Using equation (2) we can solve it for the final velocity and get the formula
(3)
v = v 0 + at
The position of the object moving with acceleration is given by the formula
1
x = x0 + v0 t + at 2 ,
(4)
2
where x0 and v0 are the initial position and velocity.
There are two other important formulas for motion with constant acceleration:
1
(5)
x = (v0 + v)t
2
and
v 2 = v 02 + 2ax
(6)
Equation (5) is based on the average velocity equals the mean of initial v0 and final v
velocities of linear motion with constant acceleration. Equation (6) can be derived from
(3) and (5) by eliminating t.
A common example of motion with constant acceleration is free fall of an object.
In absence of resistance, the free fall acceleration of any object regardless of its mass,
composition, and size is the same at the same location. This acceleration is called the
acceleration due to gravity and is a vector g directed vertically downward. The magnitude
of the gravity acceleration slightly decreases with increasing altitude and varies with
latitude. In the New York City the standard value of the gravity acceleration is g = 9.80
m/s2.
1
In the absence of friction an object moves along an inclined plane with
acceleration a, which is the component of the gravity acceleration along the plane. If the
plane is inclined to the horizontal at the angle θ, the acceleration is
a = g sin θ
(7)
a
g
θ
Two-dimensional motion is considered on the basic principle of the independent
motions in each direction. In the absence of air resistance, the projectile dose not slow
down in the horizontal direction, so it moves with constant velocity. In the vertical
direction, the projectile experiences the acceleration g. For the motion on the inclined
plane, a two-dimension motion experiences the constant acceleration a along the
direction of the inclination and no acceleration in the perpendicular direction.
The experiment is performed with the air table, which is essentially a box with
many small holes in the top surface. A blower sends air through the holes and the puck
floats on those small air fountains, which reduces the friction. The webcam makes
pictures of the top surface of the air table with the puck in equidistant intervals of time, so
we can see the sequential position of the puck. The analysis of the puck positions gives us
the ability to study its motion.
Procedure
Before starting the experiment, make sure that the air table is level. To do this put
the puck in the center of the table and turn on the blower. The puck should remain
motionless or moves very slowly. Level the table using the adjustment screws if
necessary. Turn off the blower.
The computer should already be turned on. Check with your instructor if it is not.
Double click on the icon Logger Pro on the control panel. The window Logger Pro
shows up. You will see an empty graph; a window to the left will be a blank table.
Part 1. The puck sliding downward freely
1. Put the rectangular wooden block under the leg at the center of the table edge,
using the smallest size of the block. Write down the distance between the legs and
the height of the block on the data sheet.
2
2. Put the meter stick with two 0.5 m separated marks along the edge of the table.
This will be used to establish the scale for the video pictures. Put the puck edge
under the border string in the middle of the higher side of the table and turn on the
blower.
3. In the Logger Pro window click Insert → Video Capture and two windows:
Video Capture and Logitech Quick Cam show up. The window Video Capture
should show the image of the whole top surface of the air table. If the air table
doesn’t fit to the window, ask the instructor to assist. The window Logitech
Quick Cam for setting of the webcam isn’t used in the experiment.
4. Turn on the blower. Click on Start Capture and when it is replaced by Stop
Capture gently lift the border string to release the puck. The process of taking
pictures and creation a movie of sliding down the puck is going on. Click Stop
Capture when the puck hits the lower border string. Turn off the blower and
close the Video Capture window.
5. A small frame of the table with the puck is seen in the Logger Pro window.
Enlarge the frame to the maximal size by dragging by the center or corner of it.
The icons: ► , ■ , ▐◄ , ►►,◄◄ on the bottom of the frame mean “play”,
“stop”, “ move to the first frame”, “move forward frame by frame”, “move
backward frame by frame”, respectively. You may play the video once or more to
see what you have. Set the video at the frame where you wish to start analysis.
6. Click on the icon ○◦○► Enable/Disable video analysis and the column of tool
icons shows up at the right side of the frame. Click on the Set Scale icon (fourth
from the top), coincide the cursor on the frame with a mark on the meter stick,
hold the mouse button, drag it to the other mark, and release the button. The
window Scale shows up. Type 0.5 (it is distance 0.5 m between the marks), and
click OK. The scale is set.
7. Click on the second from the top tool Add Point. The cursor looks like +. Place it
in the center of the puck in the first frame and click on it to record its coordinates.
The image moves to the next frame. Again place the cursor to the center of the
puck and click. Continue to do this until you have recorded all the frames you
wish to include in your analysis, i.e. stop before the puck hits the bottom border
string. The trajectory of the puck is marked on the frame. The current Time, X, Y
coordinate, as well as X and Y components of velocity, are recorded in the table.
8. To set the origin of the graph click on the third tool Set Origin. Move the cursor
on the point, which you choose as the origin. It could be the first point, to do this
move ◄◄to the first frame and click cursor on the center of the puck. Click on
the graph area under the frame and the graph position X and Y vs. Time shows
up.
9. Now you can begin analysis of the graph. Right click on the X and Y axis or on
the top of the window Experiment →Graph Options →Axes Options and
choose X and X-Velocity in Y-Axis Columns, and click Done. The graphs
position X and X-Velocity vs. Time are presented. If the Autoscale is not good,
go back to Graph Options →Axes Options, use Scaling →Manual (it could be
from -1 to +1), and click Done. Highlight the appropriate area for analyses of the
graph; exclude border points, which don’t fit to all other. Go to the menu Analyze
at the top of the window, select Linear for velocity or Curve Fit for position, and
3
in the appeared window choose appropriate value. Choose quadratic function for
position graph in Curve Fit window, click Try, check the line coincides with
experimental points, and click OK. Next to the graphs appear boxes with
equations and coefficients. The slope of the velocity graph m is the acceleration,
where as the coefficient A of the position graph equals to the half of the
acceleration. Write down the value of the accelerations a to the data sheet.
10. Go to File →Print Graph; better use Print Setup landscape paper orientation;
preliminarily type your surnames and comments like the used height of the
wooden block in the window Printing Options. Print the graph.
Part 2. Moving the puck up and down
Put the rectangular wooden block under the leg at the center of the table edge,
using the other, intermediate, size of the block. Write down the using height of the block
on the data sheet.
Follow the manual of Part 1 with the change to click Stop Capture after the
second time the puck hits the lower border string. For the analysis use the first frame after
the first time the puck hits the string.
Part 3. Two-dimensional motion of the puck
Put the rectangle wooden block under the leg at the center of the table edge, using
the largest size of the block. Write down the using height of the block to the data sheet.
Practice launching the puck with different velocities until you can launch it
reliably with an initial velocity such that it will follow a parabolic trajectory covering a
significant fraction of the air table. It should reach the peak of its about ¾ of the way up
the table and it should travel at least half the width of the table. Capture this motion and
analyze as in Part 1.
Print the graphs: 1. Position X and X-Velocity vs. Time, 2. Position Y and YVelocity vs. Time, and analyze the graphs. For Position Y find the slope and for YVelocity take mean value to find the average value of the velocity vy. For Part 3 print
both graphs and the table too.
Analyses and calculation
1. What would the graph of a(t) look like? Sketch it.
2. Calculate the sine for each part and the value of gravity acceleration. Compare the
average experimental value gravity acceleration to standard g = 9.80 m/s2,
calculate percent error.
3. Discuss the graph x(t) and v(t) and the significance of the parameters obtained by
the fitting program. State what each of these parameters represents.
4. Calculate the time between the frames and how many frames made in one second
using the table of the Part 3.
5. For Part 3 using the data of the table and formulas (1) and (2) calculate X and Y
components of initial velocity and X component of the acceleration. On the basis
of the calculated values and using the equation of motion, calculate the puck
trajectory maximum height, the time reach the highest point, and the range.
Compare the values to the correspondent of the graphs. All calculation must be in
the report.
4
Questions
1. Is the acceleration of the puck at the upper point of Part 2 equal to zero? If not,
why not.
2. Why might be the experimental gravity acceleration better agreement for lager,
rather than smaller, angle inclination of the air table θ?
3. Derive equation (6).
5
Student's surname _____________________
Date __________
MOTION WITH CONSTANT ACCELERATION
Distance between the air table legs = ____________ m
Part #
Block
height, m
sin θ
a from
position, m/s2
a from
velocity, m/s2
a average,
m/s2
1
2
3
Experimental gravity acceleration
% error
Part 3. Two-dimensional motion of the puck
Time between frames = ______________ s
Number frames per sec. = ______________1/s
Calculated
Initial velocity X component, m/s
Initial velocity Y component, m/s
Acceleration X component, m/s2
Time reach maximum height, s
Maximum height, m
Range of the trajectory
6
Graph
% diff
g,
m/s2