Description
Kinematics Carolina Distance Learning Investigation Manual
Unformatted Attachment Preview
Kinematics
Student Name
Date
Activity 1: Graph and interpret motion data of a moving object
Activity 1. Table 1
Time (x axis) (seconds)
Position (y axis) (meters)
0
0
5
20
10
40
15
50
20
55
30
60
35
70
40
70
45
70
50
55
Insert your graph here for Distance vs Time of a Moving Object.
Questions for Activity 1
Question 1:
What is the average speed of the train during the time interval from 0 s to 10 s?
Question 2:
2− 1
Using the equation: = 2− 1, calculate the average speed of the train as it moves from
position x = 50m to x = 60m.
Question 3:
What does the slope of the line during each time interval represent?
Question 4:
From time t = 35 s until t = 45 s, the train is located at the same position. What is slope of
the line while the train is stationary?
Question 5:
Calculate the average speed of the train as it moves from position x = 70m to x = 55m.
What does the sign of the average velocity during this time interval represent?
Question 6:
What is the displacement of the train from time t = 0s until t = 50s?
Question 7:
What is the total distance traveled by the train from time t = 0s until t = 50s?
Question 8. What is the slope of the line during the time interval t = 45 to t = 50?
Question 9: What does the sign of the slope in question 8 represent in terms of the
motion of the train?
Question 10: What is the average velocity of the train during the interval t= 0s to t =
50s?
Question 11: Does the train’s average velocity during the interval t= 0s to t = 50 s
provide a complete picture of the train’s motion during this time?
Activity 2. Calculate the velocity of a moving object.
Activity 2. Table 1
Time (s)
Displacement
(m)*
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
*Note that 0.25 m = 25 cm
Insert a graph of Table 1 here. Include a chart title, axes titles and units.
Activity 2. Table 2
Time (s)
Velocity
(m/s)
1
2
3
4
5
6
7
8
Insert a graph of Table 2 here. Include a chart title, axes titles and units.
Activity 3 Graphing the motion of an Object with Constant Acceleration
Activity 3. Data Table 1.
Time (s)
Average Time (s)
Average Time2(s2)
Distance (m)
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
*Note that 0.10 m = 10 cm
Insert your graphs of Distance vs Time (m) and Distance vs Time Squared here:
Questions for Activity 3
Question 1: What is the shape of the graph when displacement is graphed vs. time?
Question 2: What is the shape of the graph when displacement is graphed against time
squared?
Question 3: What do the shapes of these graphs tell you about the relationship
between distance and displacement for an object traveling at a constant
acceleration?
Activity 4: Predict the time for a steel sphere to roll down an incline.
A Length of Track (cm) (s)
Steel Sphere
Acrylic Sphere
80 cm
80 cm
(Step 1, use 80 cm)
B
Angle of Elevation () in Degrees ⁰
(Step 1)
C Calculated Time from s=0 to s=80
(formula from step 2)
D Measured Time from s=0 to s=80
(step 3 with stopwatch)
E
% Difference (step 4)
Question for Activity 4: What effect does the type of the sphere have on the
time of the object to travel the measured distance, explain?
Activity 5: Demonstrate that a sphere rolling down the incline is moving under
constant acceleration.
Questions for Activity 5:
1. Describe your observations of the sounds made as the sphere crosses the
equally spaced rubber bands (procedure step 4)? (If the sounds are too
fast to discern, lower the angle of the ramp.)
2. Describe your observations of the sounds made as the sphere crosses the
unequally spaced rubber bands (procedure step 9)? (Use same angle as
step4).
3. Explain the differences you observed if any between the sounds with
equal spacing and sounds with unequal spacing.
Kinematics
Carolina Distance Learning
Investigation Manual
2
Table of Contents
Overview …………………………………………………………………………………………… 3
Objectives …………………………………………………………………………………………. 3
Time Requirements ……………………………………………………………………………. 3
Background ………………………………………………………………………………………. 4
Materials ……………………………………………………………………………………………. 8
Safety ………………………………………………………………………………………………… 9
Alternate Methods for Collecting Data using Digital Devices. ……….. 10
Preparation ……………………………………………………………………………………… 11
Activity 1: Graph and interpret motion data of a moving object ….. 11
Activity 2: Calculate the velocity of a moving object ……………………. 12
Activity 3: Graph the motion of an object traveling under constant
acceleration……………………………………………………………………………………. 16
Activity 4: Predict the time for a steel sphere to roll down an incline 23
Activity 5: Demonstrate that a sphere rolling down the incline is
moving under constant acceleration …………………………………………….. 26
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Overview
Kinematics is the branch of physics that deals with the analysis of the motion of objects
without concern for the forces causing the motion. Scientists have developed
equations that describe the movement of objects within certain parameters, such as
objects moving with a constant velocity or a constant acceleration. Using these
equations, the future position and velocity of an object can be predicted. This
investigation will focus on objects moving with a constant velocity or a constant
acceleration. Data will be collected on these objects, and the motion of the objects
will be analyzed through graphing these data.
Objectives
Explain linear motion for objects traveling with a constant velocity or constant
acceleration
Utilize vector quantities such as displacement and acceleration, and scalar
quantities such as distance and speed.
Analyze graphs that depict the motion of objects moving at a constant velocity
or constant acceleration.
Use equations of motion to analyze and predict the motion of objects moving at
a constant velocity or constant acceleration.
Time Requirements
Preparation…………………………………………………………………………………5 minutes
Activity 1 …………………………………………………………………………………….15 minutes
Activity 2 …………………………………………………………………………………….20 minutes
Activity 3 …………………………………………………………………………………….20 minutes
Activity 4 …………………………………………………………………………………….10 minutes
Activity 5 …………………………………………………………………………………….20 minutes
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Background
Mechanics is the branch of physics that that studies the motion of objects and the
forces and energies that affect those motions. Classical Mechanics refers to the motion
of objects that are large compared to subatomic particles and slow compared to the
speed of light. The effects of quantum mechanics and relativity are negligible in
classical mechanics. Most objects and forces encountered in daily life can be
described by classical mechanics, such as the motion of a baseball, a train, or even a
bullet or the planets. Engineers and other scientists apply the principles of physics in
many scenarios. Physicists and engineers often collect data about an object and use
graphs of the data to describe the motion of objects.
Kinematics is a specific branch of mechanics that describes the motion of objects
without reference to the forces causing the motion. Examples of kinematics include
describing the motion of a race car moving on a track or an apple falling from a tree,
but only in terms of the object’s position, velocity, acceleration, and time without
describing the force from the engine of the car, the friction between the tires and the
track, or the gravity pulling the apple. For example, it is possible to predict the time it
would take for an object dropped from the roof of a building to fall to the ground using
the following kinematics equation:
=
1
2
2
Where s is the displacement from the starting position at a given time, a is the
acceleration of the object, and t is the time after the object is dropped. The equation
does not include any variables for the forces acting on the object or the mass or energy
of the object. As long as the some initial conditions are known, such an object’s
position, acceleration, and velocity at a given time, the motion or position of the object
at any future or previous time can be calculated by applying kinematics. This method
has many useful applications. One could calculate the path of a projectile such as a
golf ball or artillery shell, the time or distance for a decelerating object to come to rest,
or the speed an object would be traveling after falling a given distance.
Early scientists such as Galileo Galilee (1564-1642), Isaac Newton (1642-1746) and
Johannes Kepler (1571-1630) studied the motion of objects and developed
mathematical laws to describe and predict their motion. Until the late sixteenth
century, the idea that heavier objects fell faster than lighter objects was widely
accepted. This idea had been proposed by the Greek philosopher Aristotle, who lived
around the third century B.C. Because the idea seemed to be supported by
experience, it was generally accepted. A person watching a feather and a hammer
dropped simultaneously from the same height would certainly observe the hammer
falling faster than the feather. According to legend, Galileo Galilee, an Italian physicist
and mathematician, disproved this idea in a dramatic demonstration by dropping
objects of different mass from the tower of Pisa to demonstrate that they fell at the
same rate. In later experiments, Galileo rolled spheres down inclined planes to slow
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down the motion and get more accurate data. By analyzing the ordinary motion of
objects and graphing the results, it is possible to derive some simple equations that
predict their motion.
To study the motion of objects, a few definitions should be established. A vector refers
to a number with a direction and magnitude (or size). Numbers that have a magnitude
but not a direction are referred to as a scalar. In kinematics, vectors are important,
because the goal is to calculate the location and direction of movement of the object
at any time in the future or past. For example if an object is described as being 100
miles from a given position traveling at a speed of 50 miles per hour, that could mean
the object will reach the position in 2 hours. It could also mean the object could be
located up to 100 miles farther away in 1 hour, or somewhere between 100 and 200
miles away depending on the direction. The quantity speed, which refers to the rate of
change in position of an object, is a scalar quantity because no direction of travel is
defined. The quantity velocity, which refers to both the speed and direction of an
object, is a vector quantity.
Distance, or the amount of space between two objects, is a scalar quantity.
Displacement, which is distance in a given direction, is a vector quantity. If a bus
travels from Washington D.C. to New York City, the distance the bus traveled is
approximately 230 miles. The displacement of the bus is (roughly) 230 miles North-East.
If the bus travels from D.C to New York and back, the distance traveled is roughly 460
miles, but the displacement is zero because the bus begins and ends at the same point.
It is important to define the units of scalar and vector quantities when studying
mechanics. A person giving directions from Washington D.C. to New York might
describe the distance as being approximately 4 hours. This may be close to the actual
travel time, but this does not indicate actual distance.
To illustrate the difference between distance and displacement, consider the following
diagrams in Figures 1-3.
Consider the number line in Figure 1. The displacement from zero represented by the
arrowhead on the number line is -3, indicating both direction and magnitude. The
distance from zero indicated by the point on the number line equals three, which is the
magnitude of the displacement. For motion in one dimension, the + or‒ sign is sufficient
to represent the direction of the vector.
Figure 1.
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Figure 2
Figure 3
The arrows in Figures 2 and 3 represent displacement vectors for an object. The long
lines represent a displacement with a magnitude of five. This displacement vector can
be resolved into two component vectors along the x and y axes. In all four diagrams
the object is moved some distance in either the positive or negative x direction, and
then some distance in the positive y direction; however, the final position of the object
is different in each diagram. The total distance between the object’s initial and final
position in each instance is 5 meters, however to describe the displacement, s, from the
initial position more information is needed.
In Figure 2, the displacement vector can be given by 5 meters (m) at 53.1°. This vector
is found by vector addition of the two component vectors, 3 m at 0° and 4 m at 90°,
using conventional polar coordinates that assign 0° to the positive x direction and
progress counterclockwise towards 360°. The displacement in Figure 3 is 5 m at 143.1°.
In each case the magnitude of the vector is length of the arrow, that is, the distance
that the object travels. Most texts will indicate that a variable represents a vector
quantity by placing an arrow over the variable or placing the variable in bold.
To indicate the magnitude of a vector, absolute value bars are used. For example the
magnitude of the displacement vector in each diagram is 5 m. In Figure 2 the
displacement is given by:
s = 5 m at 51.3°
The magnitude of this vector may be written as:
| |=d=5m
The displacement vector in Figure 2, s = 5 m at 53.1°, can be resolved into the
component vectors 3 m at 0° and 4 m at 90°.
Two more terms that are critical for the study of kinematics are velocity and
acceleration. Both terms are vector quantities.
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Velocity (v) is defined as the rate of change of the position of an object. For an object
moving in the x direction, the magnitude of the velocity (speed) may be described as:
=
2 − 1
∆
Where x2 is the position at time t2 and x1 is the position of the object at time t1. The
variable ∆t represents the time interval t2 -t1. The symbol, ∆, is the Greek symbol delta,
and refers to a change or difference. ∆t is read, “delta t”. Time in the following
examples is provided in seconds (s). Please be sure that you do not confuse the “s” unit
for seconds, and the “s “ unit for displacement in these formulas.
For example if an object is located at a position designated x1 = 2 m and moves to
position x2 = 8 m over a time interval ∆t = 2 s, then the average speed could be
calculated:
8 − 2
= 3 /s
2s
The velocity could for this object could be indicated as:
= 3 /s
Because velocity is a vector quantity, the positive sign indicates that the object was
traveling in the positive x direction, at a speed of 3 m/s.
Acceleration is defined as the rate of change of velocity. The magnitude of
acceleration may be described as:
=
− 1
∆
For example, an object with an initial velocity v1 = 10 m/s slows to a final velocity of v2 =
1 m/s over an interval of 3 s.
1 ⁄s − 10 ⁄
= −3 ⁄s⁄
3s
s
The object has an average acceleration of ‒3 meters per second per second, which
can also be written as ‒3 meters per second squared, or ‒3 ⁄ 2 .
s
Because only the initial and final positions or velocities over a given time interval are
used in these equations, the calculated values indicate the average velocity or
acceleration. Calculating the instantaneous velocity or acceleration of an object
requires the application of calculus. Only average velocity and acceleration are
considered in this investigation.
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Materials
Included in the Central Materials kit:
Tape Measure
Rubber Bands
Protractor
Included in the Mechanics Module materials kit
Constant Velocity Vehicle
Steel Sphere
Acrylic Sphere
Angle Bar
Foam Board
Block of Clay
Needed, but not supplied:
Scientific or Graphing Calculator
or Computer with Spreadsheet Software
Permanent Marker
Masking Tape
Stopwatch, or smartphone able to record
video
Reorder Information: Replacement supplies for the Kinematics investigation can be
ordered from Carolina Biological Supply Company, kit 580404 Mechanics Module.
Call 1-800-334-5551 to order.
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Safety
Safety goggles should be worn while conducting this investigation.
Read all the instructions for this laboratory activity before beginning. Follow the
instructions closely and observe established laboratory safety practices.
Do not eat, drink, or chew gum while performing this activity. Wash your hands with
soap and water before and after performing the activity. Clean up the work area
with soap and water after completing the investigation. Keep pets and children
away from lab materials and equipment.
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Alternate Methods for Collecting Data using Digital Devices.
Much of the uncertainty in these experiments arises from human error in measuring the
times of events. Some of the time intervals are very short, which increases the effect of
human error due to reaction time.
Observing the experiment from a good vantage point that removes parallax errors and
recording measurements for multiple trials helps to minimize error, but using a digital
device as an alternate method of data collection may further minimize error.
Many digital devices, smart phones, tablets, etc. have cameras and software that
allow the user to pause or slow down the video.
If you film the experiment against a scale, such as a tape measure, you can use your
video playback program to record position and time data for the carts. This can
provide more accurate data and may eliminate the need for multiple trials.
If the time on your device’s playback program is not sufficiently accurate, some
additional apps may be available for download.
Another option is to upload the video to your computer. Different video playback
programs may come with your operating system or software suite or may be available
for download.
Some apps for mobile devices and computer programs available for download are
listed below, with notes about their features.
Hudl Technique: http://get.hudl.com/products/technique/
iPhone/iPad and Android
FREE
Measures times to the hundredth-second with slow motion features
QuickTime http://www.apple.com/quicktime/download/
Free
Install on computer
30 frames per second
Has auto scrubbing capability
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Preparation
1. Collect materials needed for this investigation.
2. Locate and clear an area of level floor space in order to conduct the constant
velocity experiment. The space should be free of obstruction and three to four
meters long with a surface which will allow the vehicle to maintain traction but not
impede the vehicle.
Activity 1: Graph and interpret motion data of a moving object
One way to analyze the motion of an object is to graph the position and time data.
The graph of an object’s motion can be interpreted and used to predict the object’s
position at a future time or calculate an object’s position at a previous time.
Table 1 represents the position of a train on a track. The train can only move in one
dimension, either forward (the positive x direction) or in reverse (the negative x
direction).
Table 1
Time (x-axis), seconds
0
5
10
15
20
30
35
40
45
50
Position (y-axis), meters
0
20
40
50
55
60
70
70
70
55
1. Plot the data from Table 1 on a graph using the y-axis to represent the displacement
from the starting position (y = 0) and the time coordinate on the x-axis.
2. Connect all the coordinates on the graph with straight lines.
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Activity 2: Calculate the velocity of a moving object
In this activity you will graph the motion of an object moving with a constant velocity.
The speed of the object can be calculated by allowing the Constant Velocity Vehicle
to travel a given distance and measuring the time that it took to move this distance. As
seen in Activity 1, this measurement will only provide the average speed. In this activity,
you will collect time data at several travel distances, plot these data, and analyze the
graph
1. Find and clear a straight path approximately two meters long.
2. Install the batteries and test the vehicle.
Note: The vehicle should be able to travel two meters in a generally straight path. If
the vehicle veers significantly to one side, you may need to allow the vehicle to
travel next to a wall. The friction will affect the vehicle’s speed, but the effect will be
uniform for each trial.
3. Use your tape measure or ruler to measure a track two meters long. The track should
be level and smooth with no obstructions. Make sure the surface of the track
provides enough traction for the wheels to turn without slipping.
Place masking tape across the track at 25 cm intervals.
4. Set the car on the floor approximately 5 cm behind the start point of the track.
Note: Starting the car a short distance before the start point allows the vehicle to
reach its top speed before the time starts and prevents the short period of
acceleration from affecting the data.
5. Set the stopwatch to the timing mode and reset the time to zero.
6. Start the car and allow the car to move along the track.
7. Start the stopwatch when the front edge of the car crosses the start point.
8. Stop the stopwatch when the front edge of the car crosses the first 25 cm point.
9. Recover the car, and switch the power off. Record the time and vehicle position on
the data table.
10. Repeat steps #5‒9 for each 25 cm interval marked. Each trial will have a distance
that is 25 cm longer than the previous trial, and the stopwatch will record the time
for the car to travel the individual trial distance.
11. Record the data in Data Table 1.
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Data Table 1
Time (s)
Displacement (m)
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
12. Graph the time and displacement data points on graph paper.
13. Draw a line of best fit through the data points.
Note: The points should generally fall in a straight line. If you have access to a
graphing calculator or a computer with spreadsheet software, the calculator or
spreadsheet can be programmed to draw the line of best fit, or trend line.
14. Calculate the slope of the line.
Note: Based on the equation of a line that cross the y-axis at y = 0, the slope of the
line, m, will be the velocity of the object.
=
= ∆
15. Make a second data table, indicating the velocity of the car at any time.
Data Table 2
Time (s)
1
2
3
4
5
6
7
8
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Velocity (m/s)
14
Note: Because the object in this example, the battery-powered car, moves with a
constant speed, all the values for the velocity of the car in the second table should
be the same. The value of the velocity for the car should be the slope of the line in
the previous graph.
16. Graph the data points from the Data Table 2 on a second sheet of graph paper.
Label the y-axis Velocity and the x-axis Time.
Note: When the data points from this table are plotted on the second graph, the
motion of the car should generate a horizontal line. On a velocity vs. time graph, an
object moving with a constant speed is represented by a horizontal line.
17. Draw a vertical line from the x-axis at the point time = 2 seconds so that it intersects
the line representing the velocity of the car.
18. Draw a second vertical line from the x-axis at the point time = 4 seconds so that it
intersects the line representing the velocity of the car.
19. Calculate the area represented by the rectangle enclosed by the two vertical lines
you just drew, the line for the velocity of the car, and the x-axis. An example is shown
as the blue shaded area in Figure 4.
Figure 4
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Note: In order to calculate the area of this rectangle, you must multiply the value
for the time interval between time t=2 s and time t=4 s, by the velocity of the car.
This area represents the distance traveled by the object during this time interval.
This technique is often referred to as calculating the “area under the curve”. The
graph of velocity vs. time for an object that is traveling with a constant
acceleration will not be a horizontal line, but using the same method of graphing
the velocity vs. time and finding the “area under the curve” in a given time
interval can allow the distance traveled by the object to be calculated.
Distance = velocity × time
In this equation, the time units (s) cancel out when velocity and time are
multiplied, leaving the distance unit in meters.
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Activity 3: Graph the motion of an object traveling under constant
acceleration
Collecting data on freefalling objects requires accurate timing instruments or access to
a building with heights of several meters where objects can safely be dropped over
heights large enough to allow accurate measurement with a stopwatch. To collect
usable data, in this activity you will record the time objects to roll down an incline. This
reduces acceleration to make it easier to record accurate data on the distance that
an object moves.
1. Collect the following materials:
Steel Sphere
Acrylic Sphere
Angle Bar
Clay
Tape Measure
Timing Device
Protractor
2. Use the permanent marker and the tape measure to mark the inside of the angle
bar at 1-cm increments.
3. Use the piece of clay and the protractor to set up the angle bar at an incline
between 5° to 10°. Use the clay to set the higher end of the anglebar and to
stabilize the system. (Figure 5)
Figure 5
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Set up the angle bar so that the lower end terminates against a book or a wall, to stop
the motion of the sphere (Figure 6.)
Figure 6
4. Place the steel sphere 10 cm from the lower end of the track.
5. Release the steel sphere and record the time it takes for the sphere to reach the
end of the track.
6. Repeat steps #4‒5 two more times for a total of three measurements at a starting
point of 10 cm.
7. Repeat steps #4‒6, increasing the distance between the starting point and the end
of the track by 10 cm each time.
8. Record your data in Data Table 3.
Note: You are recording the time it takes for the sphere to accelerate over an
increasing distance. Take three measurements for each distance, and average the
time for that distance. Record the time for each attempt and the average time in
Table 4.
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Data Table 3
Time (s)
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Trial 1 =
Trial 2 =
Trial 3 =
Average time (s)
Average Time 2 (s2)
Distance (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
9. Calculate the average time for each distance and record this value in Table 4.
10. Create a graph of distance vs. time using the data from Table 4.
11. Complete Table 4 by calculating the square of the average time for each distance.
12. Create a graph of displacement vs. time squared from the data in Table 4.
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Graphing the displacement vs time data from Table 4 will generate a parabola.
When data points generate a parabola, it means the y value is proportional to the
square of the x value, or:
∝ 2
That means the equation for a line that fits all the data points looks like:
= 2 + + .
In our experiment, the y-axis is displacement and the x-axis is time-; therefore
displacement is proportional to the time squared:
∝ 2
So, we can exchange y in the equation with displacement (s), to give a formula that
looks like:
= 2 + + .
We would know the displacement s, at any time t. We just need to find the
constants, A, B, and C.
The equation that describes the displacement of an object moving
with a constant acceleration is one of the kinematics equations:
1
= ∆ 2 + ∆
2
The following section describes how to find this equation using the same method of
finding the “area under the curve” covered in Activity 2.
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Finding an Equation for the Motion of an Object with Constant Acceleration
The general form of a line is:
= +
Where m is the slope of the line, and b is the y-intercept, the point where the line
crosses the y-axis. Because the first data point represents time zero and
displacement zero, the y-intercept is zero and the equation for the line simplifies
to:
y = mx
The data collected in Activity 3 showed that:
∝ 2
This means that the displacement for the object that rolls down an inclined plane
is can be represented mathematically as:
= 2 + c
Where k is an unknown constant representing the slope of the line, and c is an
unknown constant representing the y-intercept.
The displacement of the sphere as it rolls down the incline can be calculated
using this equation, if the constants k and c can be found.
Further experimentation indicates that the constant k for an object in freefall is
one-half the acceleration. If the object is released from rest, the constant c will
be zero.
So for an object that is released from rest, falling under the constant
acceleration due to gravity, the displacement from the point of release is given
by:
1
= 2
2
Where s is the displacement, t is the time of freefall, and is the acceleration.
For objects in freefall near Earth’s Surface the acceleration due to gravity has a
value of 9.8 ⁄ 2 .
s
Another way to derive this equation, and find the values for k and c, is to
consider the velocity vs. time graph for an object moving with a constant
acceleration. Remember the velocity vs. time graph for the object moving with
constant velocity from Activity 2. If velocity is constant, the equation of that
graph would be:
=
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Where v represents the velocity, plotted on the y-axis, and k is the constant
value of the velocity. Plotted against time on the x-axis, this graph is a horizontal
line, as depicted in Figure 7.
Figure 7
By definition, the shaded area is the distance traveled by the object during the
time interval:
Δ = 2 − 1
=
=
∆
∴ = ∆
If an object has a constant acceleration, then by definition:
=
Or :
− 1
∆
2 = ∆ +
This is equation is in the general form of a line y = mx + b, with velocity on the yaxis and time on the x-axis. The graph of this equation would look like the graph
in Figure 8.
Figure 8
Similar to how the shaded area A1 in Figure 7 represents the distance traveled by
the object during the time interval Δt = t2 – t1, the shaded area A2 combined with
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A1 equals the distance traveled by the object undergoing constant
acceleration.
The area A1 can be given by:
1 = ∆
The area A2 can be given by:
2 =
1
( − )∆
2 2
Because this is the area of the triangle, where the length of the base is Δt and the
height of the triangle is ( − ),
Adding these two expressions and rearranging:
=
And substituting:
1
( − )∆
2 2
2 = ∆ +
Gives this equation:
=
1
( Δ + 1 + 2 Δ + 1 Δ )
2
Simplifying gives:
1
= ∆ 2 + ∆
2
This equation gives the theoretical displacement for an object undergoing a
constant acceleration, , at any time t, where s is the displacement during the
time interval, Δ , and v1 is the initial v
