Kinematics of Freefall
Introduction
The purpose of this experiment is to study the motion of an object in free fall, and to deduce from the
measurements, a value for the acceleration due to gravity, g, which we will compare to the accepted value
of 980.35 cm/s2.
Materials and Equipment:
• Freefall tower apparatus
• High voltage spark generator
• Paraffin wax paper
•
•
Two-meter measuring stick
Masking tape
Theory:
According to the general theory of motion, the vertical position, y, and velocity, v, of an object falling
freely with constant acceleration of magnitude g, are given by:
𝑎=𝑔
(1)
𝑣 = 𝑣) + 𝑔𝑡
(2)
𝑦=
.
𝑔𝑡 /
/
+ 𝑣) 𝑡 + 𝑦)
(3)
In using the above equations for this experiment, the following is assumed:
1. The vertical position, y, the velocity, v, and the acceleration, g, are measured positively in the
downward direction.
2. At time t = 0 sec:
a. y = y0 = 0 cm
b. v = v0
If both sides of Eq. (3) are divided by t, and y0 is removed from the expression, the following equation is
obtained:
𝑦
=
𝑡
.
𝑔𝑡
/
+ 𝑣)
(4)
Plotting y/t versus t results in a straight line with slope equal to ½ g and intercept v0. This equation forms
the basis for the data analysis for this experiment.
This is the first introduction of linearizing data. The 2DStats macro will not be useful determining g from
the initial measurements of y vs t. Instead, the appropriate equation is rearranged using algebra to put it
in a familiar, linear form (y = m + b). 2DStats can now be used to find a linear relationship between
these linearized variables. Linearization is a particularly useful and powerful tool which allows
measurements of exponential, trigonometric, and other non-linear relationships to be arranged in a
simpler, more accessible form.
Physics 181 – UMass Boston
Apparatus:
The acceleration of gravity, on a freely falling body, is too large to allow for more than crude
measurements using a manually operated timer. Therefore, this experiment utilizes a special mechanism
(Fig. 1) which automatically records the position of a falling body at intervals of 1/60th second.
The apparatus consists of two main components:
•
An electromagnet used to suspend, and
control the release of the falling object.
The electromagnet is connected to a DC
power supply through a trigger. When the
trigger is pressed current through the
electromagnet is cut-off, causing a loss of
the magnetic field and the release of the
object which then falls freely under the
influence of gravity.
•
A sparking device that is connected to a
high voltage source. It sparks at a known
frequency of 60 Hz (60 sparks per
second). Sparks travel from the outer wire
(marked in orange in the figure), through
a metal ring on the falling object, through
a waxed paper tape, and onto a second
wire (in black). The apparatus is arranged
such that the metal ring does not touch
either wire and so short pulses of high
voltage electricity travel between the
components as sparks.
Figure 1: Freefall apparatus
Sparks will travel from one wire to the other at the location of the
metal ring on the falling object. As this object falls between the wires
each spark will leave a mark on the wax paper tape, thus recording the
position of the object at fixed time intervals during its fall. The space
between successive marks on the paper represents the distance the
object travelled in 1/60th second. Note that the distance between marks
increases, this is a result of the object accelerating.
Figure 2: Schematic representation of generated
sparks due to induction
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Physics 181 – UMass Boston
Procedure:
1. The falling object (bob) is held in place by an electromagnet until the experimenter is ready to make
a record. It is released by opening the switch to the electromagnet. As the body falls, its position is
marked by a small burn mark on the wax paper at time intervals of 1/60th second. When the record
of the fall has been made, the tape is removed from the apparatus.
2. Fasten the tape to the top surface of the lab table using masking tape.
3. Inspect the tape for missing marks. Caution: The sparking apparatus sometimes misses a spark, be
sure to take proper account of numbering the marks in Step 5.
4. Position the two-meter stick edgewise over the tape, and line up the 10 centimeter mark with the
sixth spot. Be sure the measuring stick is aligned with the row of marks, and is not obscuring any
data. Then secure the measuring stick in place using masking tape at several points.
5. Beginning with the sixth spot, aligned at the 10 centimeter mark, label each mark starting with
0/60. The next mark should be labelled 1/60 and so on until you reach the end of the tape.
Note: When the object passed the sixth spot it already had some non-zero velocity. We select this
spot to begin our measurement (t = 0.00 s) and that initial velocity is our v0.
6. Measure the distance from the spot marked 0/60 to each of the other spots. Remember: These
measurements are starting from the 10 centimeter mark on the measuring stick, be sure to subtract
the initial 10 centimeters from your distances.
7. Confirm all measurements before removing the measuring stick.
Analysis
Linearize your data and then run the Excel 2DStats macro to determine the slope of the linear relationship.
Determine, and properly report, the experimental value of g and the initial velocity, v0, of the object.
As always, perform a precision versus accuracy check and calculate the percent difference between your
experimental value for gexp and the accepted value.
Create a hand-drawn graph of your linearized data.
Questions
1. (6 points) Based on your hand-drawn graph is acceleration constant? How did you determine this?
2. (8 points) Using your experimental values v0 and gexp compute the instantaneous velocity v (Eq. 2) of
the falling object at t = 15.5/60 seconds.
3. (10 points)
a. Compute the average velocity on the interval t = 15/60 sec to t = 16/60 sec.
𝑣̅345 =
∆7
∆8
=
79:/:< =79>/:<
89:/:<= 89>/:<
b. Compare your answer from Question 2 to your answer from Question 3a. Are these
numbers the same? Should they be?
4. (6 points) The R2 value is an important guide when judging the relationship between two quantities. If
you performed linear 2DStats calculations on the original non-linearized data y versus t explain how
the value of R2 would help you spot such a mistake.
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