Rev. 9/10 (AJB)
One Dimensional Kinematics: Gravitational Free-Fall
Objectives:
To
study
displacement, velocity and acceleration in
free fall, and to measure the acceleration due
to gravity (g) near the Earth’s surface
Your instructor will show the
use the dropping/timing circuit. The
electromagnet is energized by a power
supply and holds the iron ball. When the
switch on the landing pad is thrown, the
timer starts as the ball drops. When the ball
hits the pad, it sends a stop signal to the
timer. Your data should consist of at least
six Δy values (all negative). For each Δy
record three Δt values to the nearest
thousandth of a second and average to
produce the average Δt corresponding to
each Δy . Be sure to include a wide range of
Δy values, from a few tenths of a meter to
several meters. Include (0,0) as one of your
data points.
Experimental
Procedure:
Steel sphere, electromagnetic
holder and timing system, meter stick.
Apparatus:
Neglecting air resistance, all
objects near the Earth, that are subject only
to the Earth’s gravity, experience the same
downward constant acceleration (symbol g).
Its accepted value is 9.80 m s 2 .
Fundamentals:
For vertical motion, an object’s location is
usually specified using a y-axis with positive
meaning up and negative, down. An object’s
displacement ( Δy ) is its change in location,
Δy = y − y0 . In this experiment the object is
dropped from rest, thus its displacement is a
downward and is a negative quantity.
Average velocity is defined as vavg = Δy Δt
(Your instructor might
suggest entering your data in columns in
software such as Excel. Excel can be used to
calculate values in other columns from
measured values entered in these data
columns as well as to create graphs and to
determine numerical fits to the data.)
Analysis and Report:
and thus is negative here.
As an object falls from rest, its velocity
increases in magnitude (speed) at a constant
rate. For constant acceleration, the average
velocity is related to the initial and final
velocities by vavg = ( vf + v0 ) 2 . Since here
v0 = 0 , then vavg = vf 2 . Hence
vf = 2vavg = 2Δy Δt
1. For each displacement, determine (using
Excel?) the final (negative) velocity.
2. First plot (Excel?) Δy (negative values
on the vertical axis) versus Δt (horizontal
axis). The graph will not be linear. Use a
curve-fitting routine, set to a polynomial
of order 2 to determine the equation of
the best fit and from Eq. 3, extract a value
for g.
(1)
Acceleration is the time rate of change of
velocity, or
(2)
a = Δv Δt
3. Next plot (Excel?) Δy (vertical axis)
versus (Δt)2 (horizontal axis). It should be
a straight line. From a linear fit and using
Eq. 3, extract a second value for g.
Thus the acceleration due to gravity is also a
negative quantity. If g is constant, then a
dropped object’s displacement will vary
quadratically as
2
(3)
Δy = ( −g 2 ) ( Δt )
(over)
(You should be able to show this result from
the constant acceleration free fall equations.)
1
F. Calculate the speed the ball would have
to leave the floor of the laboratory so that
it just barely touched the ceiling. How
long would it take to reach the ceiling?
Analysis and Report: (continued)
4. Next graph v f versus Δt and determine
the equation of the best fit to the data.
From this equation extract yet a third
experimental value for g.
G. If this experiment were performed on the
Moon where the acceleration due to
gravity is only one-sixth of the value on
Earth, how long would the object have
taken to fall its longest distance? What
would its landing speed have been?
5. In a table, report your three values of g
from each method used and the average.
It should be within +5% of the accepted
values.
.
6. List some of the important experimental
sources of uncertainty in this experiment.
H. What would the final velocity versus
time graph look like if the acceleration
due to gravity were considerably larger
near the floor of the lab compared to the
ceiling? Explain your reasoning.
(Your instructor may
or may not require some or all of these in
your report). Show your work and use your
data, graphs, and/or free-fall equations to
answer the following questions.
Follow-Up Questions:
I. What would a graph of acceleration versus
time look like in (a) your experiment, and
(b) on the Moon?
A. For the largest fall distance measured,
quantitatively compare the time the ball
took to fall the first half of that distance
to the time to fall the second half.
Explain why the first half took longer.
B. For the largest fall distance measured,
quantitatively compare the speed of the
ball when half the drop time has elapsed
to the speed just before landing. Is the
speed at the half-time point equal to half
of the final speed? Why or why not?
C. For the largest fall distance measured,
quantitatively compare the distance the
ball fell during the first half of the total
time to the distance fallen in the last
half. Explain why it fell much further
during the second half.
D. For the largest fall distance measured,
quantitatively compare the speed of the
ball when it had fallen half the distance
to its final speed. Is the halfway speed
half of the final speed? Why or why not?
E. For the longest distance measured,
suppose the total drop time was 0.200
seconds less than you measured because
the ball was given some initial downward
speed. Determine its initial speed.
2