motion of objects that have constant acceleration is common in nature, Motion with constant acceleration is known
uniform accelerated motion. The acceleration for this type of motion is defined as
Av
(5.3)
a
V₂-V
t₂-t,
vo
1
Vi are the final and initial instantaneous velocities at the instants t, and t, respectively. The acceleration is
positive if the velocity increases with time and is negative if the velocity decreases with time. To simplify our notation, let us
take the initial time t, to be zero: t = 0. This is effectively starting a stopwatch at tq. We can then let t = t, be the elapsed
Fig. 5.2. Experim
acceleratic
where
and
V2
time. The initial and final velocities will now be represented by v, and v. For these assumptions equation (3) becomes
(5.4)
Procedure
V-Vo
a=
t
We can solve this equation for v and obtain
(5.5)
V = Vo + at.
It can also be shown that the displacement x for uniformly accelerated motion at a later time t is given by
at?
(5.6)
x = Vot+
The aim here i
AL and the times
The second air
This goal can be a
cart to pass betwee
1.
Set up the dyn
2. Measure the le
3.
Connect the
Connect the D
5. Place Photog
6. Set up the se
Open Da
2
We may combine equations (5.5) and (5.6) to eliminate t and obtain a relationship for a, x, V., and v. We first solve equation
(5.5) for t and then substitute the resulting expression into equation (5.6). After simplification we finally get
v2 – vă = 2ax
(5.7)
4.
or
X=
z? - vo
(5.8)
2a
The objectives of this laboratory activity can now be achieved by determining the velocity and the acceleration according to
equations (5.2) and (5.4) and then subsequently verifying the kinematic equations (5.6) and (5.8).
The actual experimental configuration is shown in Fig. 5.1. This experiment requires two photogate timers to measure
the average velocities of a moving cart in two positions on an inclined dynamics track. A flag of known length is placed on
the top of the cart, and the photogate timers are mounted so that the flag obscures the light beam of the photogate as it passes
through. When the flag interrupts the light beam to the photocell of the photogate, the timer is turned on. When the flag
completely passes through the photogate, the timer turns off. Therefore, the photogate timer records the time At for a flag of
given length AL to pass through the photogate. Measuring the time taken for the flag of known length to pass through the
photogate and dividing the flag length by this time gives the average velocity of the cart as
Find the
result th
again, a
In the E
the initi
1, when
from th
“Unblo
Now in
need to
by the
sequer
Now i
moves
AL
V=
At
(5.9)
Photo
“Bloc
To IE
“Cho
icon
your
an
Because the cart is accelerated during the measurement, the calculated velocity is the average of the true velocity. Since we
are using the short length flag AL, the
average velocity approaches the true
Photogate 1
Flag
Photogate 2
instantaneous velocity of the cart. The
shorter the flag,
the better the
Computer
approximation.
Thus,
average
velocity can be used as an instantaneous
velocity
By using two photogate timers in
tandem and this technique of measuring
velocity, we determine the initial
velocity vo and the velocity v at any
Fig. 5.1. Setup for measurements of the velocity and acceleration of the moving
cart and verification of the kinematic equations.
later time t from equation (5.9), and, as a
result, determine the acceleration of the
7. Place the
button a
and bet
Data Ta
8. Displac
initials
34
bject
Linear Uniform Accelerated |
Motion
Lab 5
Learning Objectives
In this laboratory activity you explore the relationship between position, velocity, and acceleration for a
motion with constant acceleration. There are two objectives. The first is to measure the velocity of an
object. The second is to measure the acceleration of a moving object and verify the kinematic equations for
uniformly accelerated motion.
Required Equipment
O
Ruler
Dynamics cart track or air track
Dynamics cart or glider
Science Workshop Interface box
Two Photogate accessories
Set of 5 metallic rods of different length
Learning Outcomes
You will be able to:
Understand the concept of acceleration.
Explore the relationship between position, velocity, and acceleration.
Analyze the graphs of position and velocity versus time for an object's motion.
Determine the instantaneous velocity and acceleration of an object.
Explain how the slope of the velocity graph relates to the acceleration.
Use a computer and time sensors (photogates) for a precise measurement of time.
Use Excel for analysis and find the equation of the trend line.
Verify the kinematic equations for uniformly accelerated motion.
Theoretical Background
As an object moves, the value of its coordinate changes with time. Consider an object with the initial position's
coordinate being x, at some time t, and the final position's coordinate being x, at some later time tz. The displacement
between these two positions is defined by Ar = x2 – X;. The physical quantity that shows the rate at which the position
changes is known as the velocity. The average velocity of the object is defined as the ratio of the displacement Ar to the time
interval At:
Ax
(5.1)
V
av
=
At
where At = tą - t . Note that the displacement and the average velocity may be either positive or negative depending on
whether x, is greater or less than x . A positive value indicates motion to the right and a negative value indicates motion to
the left.
An instantaneous velocity v of the object indicates how fast the object moves and the direction of the motion at each
instant of time. The magnitude of the instantaneous velocity is called the instantaneous speed. If you measure the average
velocity of the object over smaller and smaller distances, the value of the average velocity approaches the value of the
object's instantaneous velocity, and we can write
(5.2)
Ar
V=
At
In other words, as At becomes smaller, Ar becomes smaller as well, but their ratio approaches a constant value.
When the velocity of the object is changing with time, the object is said to be accelerating. The simplest accelerated
motion is a linear motion with constant acceleration, when the velocity changes at the same rate throughout the motion. The
33
is known as
Av=V - Vo
(5.3)
Slope = a=
V – Vo
Vo
1
t
t
celeration is
tation, let us
the elapsed
Fig. 5.2. Experimental determination of the
acceleration from the slope of the graph.
cart from equation (5.4). On the other hand, analysis of equation (5.5)
shows the linear dependence between the velocity v and the elapsed time
t. Using this fact we can experimentally find acceleration of the cart by
measuring its velocity at different positions and the corresponding
elapsed time t. A plot of the velocity v versus the elapsed time t gives a
straight line that intersects the v axis at point vo, as shown in Fig. 5.2.
According to equation (5.5), the slope of this straight line is equal to the
magnitude of the acceleration. Therefore, by finding the slope from the
graph we can determine the acceleration. In this way, we experimentally
determine the acceleration.
omes
(5.4)
Procedure
(5.5)
AL and the times At, and At it takes the flag of known length to pass through the photogates.
The aim here is to determine the initial vo and final v velocities of the moving cart by measuring the length of the flag
The second aim is to determine the acceleration of the moving cart and verify the kinematic equations (5.6) and (5.8).
(5.6)
ve equation
(5.7)
(5.8)
ccording to
לל,
to measure
s placed on
as it passes
en the flag
or a flag of
hrough the
cart to pass between two photogates, and the distance x between the photogates.
1. Set up the dynamics track and the photogates as shown in Fig. 5.1.
This goal can be achieved by using the measured velocities, and w of the moving cart, the time interval t it takes for the
2. Measure the length of the flag AL and record the result in the data Data Table 5.1.
3. Connect the Digital Photogate I accessory plug into Digital Channel 1 on the Science Workshop Interface box.
4. Connect the Digital Photogate 2 accessory plug into Digital Channel 2 on the Science Workshop Interface box.
6. Set up the sensors in the software.
5. Place Photogate 2 at a distance of x=0.50 m below Photogate 1. Record this distance in Data Table 5.1.
Open DataStudio Window. Click “Create Experiment”.
Find the “Photogate” in the Sensors list in the Experiment Setup window. Double-click the “Photogate”, and as a
result the “Photogate” icon appears below Digital Channel 1 of the Interface box 750. Double-click the "Photogate”
again, and as a result the “Photogate” icon appears below Digital Channel 2 of the interface.
In the Experimental Setup Window click “Timers”, and as a result the “Timer Setup” window appears. To measure
the initial velocity V, of the cart you need to measure the time At, the cart's flag requires to pass through Photogate
1, when the Photogate 1 is first blocked by the flag and when it is unblocked as the back end of the flag moves away
from the Photogate 1. In the timing sequences choices click Ch. 1 and select "Blocked”. Click once again and select
"Unblocked".
Now in the “Timer Setup” window click +New and Timer 2 appears. To measure the final velocity v of the cart you
need to measure the time At the cart's flag requires to pass through Photogate 2, when Photogate 2 is first blocked
by the flag and when it is unblocked as the back end of the flag moves away from Photogate 2. In the timing
sequences choices click Ch. 2 and select "Blocked”. Click once again and select “Unblocked”.
Now in the "Timer Setup” window click +New again and Timer 3 appears. To measure the time t, when the cart
moves between Photogate 1 and Photogate 2 (Photogate 1 is first blocked by the flag and then the flag blocks
Photogate 2), in the timing sequences choices click Ch. 1 and select “Blocked”, then click Ch. 2 and select
“Blocked”. Finally click “Done” and all timing will be set.
To record your measurements of time, double-click the Table icon in the Display window, and as a result the
"Choose a Data Source” window appears. Choose "Timer l” and click OK. (You
can also click-and-drag the Table
icon from the Display window to the Data window for Timer 1). Repeat this procedure for Timer 2 and Timer 3. For
your measurements you only need “Elapsed time”. Click on the “Clock” icon in each table menu, and as a result you
will record just “Elapsed time”. Resize the tables to fit your screen.
(5.9)
Since we
ag AL, the
s the true
e cart. The
better the
average
tantaneous
timers in
measuring
e initial
v at any
and, as a
7. Place the cart at the starting position. The cart must be released from the same point for each run. Click the "Start"
button and release the cart from the starting point. The times Ato, At, and t the flag takes to pass through the photogates
and between the photogates will be immediately displayed. Click the "Stop” button. Record the values At,, At, and t in
8. Displace Photogate 2 by 0.1 m down and record the distance x in Data Table 5.1. Again release the cart from the same
initial starting position and repeat step 7.
Data Table 5.1.
on of the
35
.
sie
21
9. Increase the displacement of the second Photogate by an additional 10 cm to a maximum separation of 0.9 m between the
Photogates.
Computations and Data Analysis
1.
Use equation (5.9) and compute the velocities v, and v based on the length of the flag AL and measured Ato, At, and t.
2. Use equation (5.4) and compute the acceleration of the cart for each run. We assume here that vo and v are the
instantaneous velocities of the moving cart. Find the average acceleration. Record your results in Data Table 5.1.
3. Plot a graph of the velocity v, versus the time t. If you are using Excel, display the equation of the trend line on the chart
to find the slope. From the slope of the graph find the acceleration and compare this with the average acceleration by
computing the percent difference.
4. Using the experimental values of t, V., and a, determine x from equation (5.6). Compare this result with
measurement for x by computing the percent difference. Make conclusions.
5. Plot a graph of the distance x versus the time t.
6. Using the experimental values of vo , V, and a, determine x from equation (5.8).
your
Questions
1.
2.
3.
Does a speedometer in a car measure speed or velocity? Does an odometer in a car measure distance or displacement?
If the velocity of an object is zero, does its acceleration have to be zero?
A cart has a constant acceleration of a=2 m/s”. At a certain instant its speed is 5 m/s. What is its speed 2 s later and 2 s
earlier? What is its average speed during the time interval of 4 s?
2
Section
Date
Student's name
Data Table 5.1. Measurement of the acceleration of the moving cart and verification of the kinematic equations
Length of the flag AL =
m
Distance
Distance
Acceleration
% difference
for
Time
v2 – v?
Time
at2
X =
9
V-Vo
x= Vot +
Distance
between
photogates
X,
>
tama
х
At,
Ato,
2a
2
Velocity
AL
VE
At
m/s
Velocity
AL
VO
Διο
m/s
a =
Time
between
photogates
t,
S
-
t
m
S
т
m/s?
m
Average acceleration, m/s²
Slope of the graph, m/s²
Acceleration from the slope of the graph, m/s²
Percent difference for acceleration
37
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