Test of understanding graphs in kinematics v2 answers

Kinematics

Kinematics Carolina Distance Learning

Investigation Manual

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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:

𝒔 = 1

2 𝒂 𝑡2

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:

| 𝒔 | = d = 5 m

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:

𝒗 = 𝑥2 − 𝑥1

∆𝑡

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:

8𝑚 − 2𝑚

2s = 3𝑚/s

The velocity could for this object could be indicated as:

𝒗 = 3 𝑚/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:

𝒂 = 𝒗𝟐 − 𝒗1

∆𝑡

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 𝑚 s⁄ − 10

𝑚 𝑠⁄

3s = −3 𝑚 s

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 𝑚 s2⁄

.

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

http://get.hudl.com/products/technique/
http://www.apple.com/quicktime/download/
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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 Position (y-axis), meters

0 0

5 20

10 40

15 50

20 55

30 60

35 70

40 70

45 70

50 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.

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.

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.

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.

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.

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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.

14. Calculate the slope of the line.

15. Make a second data table, indicating the velocity of the car at any time.

Data Table 2

Time (s) Velocity (m/s)

1

2

3

4

5

6

7

8

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.

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. 𝑦 = 𝑚𝑥 𝑑 = 𝑣∆𝑡

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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.

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

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.

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.

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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) Average time (s) Average Time 2 (s2) Distance (m)

Trial 1 =

0.1 Trial 2 =

Trial 3 =

Trial 1 = 0.2

Trial 2 =

Trial 3 =

Trial 1 =

0.3 Trial 2 =

Trial 3 =

Trial 1 =

0.4 Trial 2 =

Trial 3 =

Trial 1 =

0.5 Trial 2 =

Trial 3 =

Trial 1 =

0.6 Trial 2 =

Trial 3 =

Trial 1 =

0.7 Trial 2 =

Trial 3 =

Trial 1 =

0.8 Trial 2 =

Trial 3 =

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:

𝒚 ∝ 𝑥2

That means the equation for a line that fits all the data points looks like:

𝑦 = 𝐴𝑥2 + 𝐵𝑥 + 𝐶.

In our experiment, the y-axis is displacement and the x-axis is time-; therefore

displacement is proportional to the time squared:

𝒔 ∝ 𝑡2

So, we can exchange y in the equation with displacement (s), to give a formula that looks like:

𝒔 = 𝐴𝑡2 + 𝐵𝑡 + 𝐶.

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:

𝒔 = 1

2 𝒂∆𝑡2 + 𝒗𝟏∆𝑡

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: 𝑦 = 𝑚𝑥 + 𝑏

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:

𝒔 ∝ 𝑡2

This means that the displacement for the object that rolls down an inclined plane

is can be represented mathematically as:

𝒔 = 𝑘𝑡2 + 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:

𝒔 = 1

2 𝒂𝑡2

Where s is the displacement, t is the time of freefall, and 𝒂 is the acceleration. For objects in freefall near Earth’s Surface the acceleration due to gravity has a

value of 9.8 𝑚 s2⁄

.

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: 𝒗 = 𝑘

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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: Δ𝑡 = 𝑡2 − 𝑡1

𝒗 = 𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕

𝑡𝑖𝑚𝑒 =

𝒔

∆𝑡

∴ 𝒔 = 𝒗∆𝑡

If an object has a constant acceleration, then by definition:

𝒂 = 𝒗𝟐 − 𝒗1

∆𝑡

Or : 𝒗2 = 𝒂∆𝑡 + 𝒗𝟏

This is equation is in the general form of a line y = mx + b, with velocity on the y-

axis 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:

𝐴1 = 𝒗𝟏∆𝑡

The area A2 can be given by:

𝐴2 = 1

2 (𝒗2 − 𝒗𝟏)∆𝑡

Because this is the area of the triangle, where the length of the base is Δt and the height of the triangle is (𝒗𝟐 − 𝒗𝟏),

Adding these two expressions and rearranging:

𝒔 = 1

2 (𝒗2 − 𝒗𝟏)∆𝑡

And substituting: 𝒗2 = 𝒂∆𝑡 + 𝒗𝟏

Gives this equation:

𝒔 = 1

2 (𝒂Δ𝑡 + 𝑣1 + 𝑣2Δ𝑡 + 𝑣1Δ𝑡)

Simplifying gives:

𝒔 = 1

2 𝒂∆𝑡2 + 𝒗𝟏∆𝑡

This equation gives the theoretical displacement for an object undergoing a

constant acceleration, 𝒂, at any time t, where s is the displacement during the time interval, Δ𝑡, and v1 is the initial velocity.

If the object is released from rest, as in our experiment, v1 = 0 and the equation simplifies to:

𝒔 = 1

2 𝒂∆𝑡2

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Activity 4: Predict the time for a steel sphere to roll down an incline

In this activity you will use the kinematics equation:

𝒔 = 1

2 𝒂∆𝑡2

This will allow you to predict how long the sphere will take to roll down the

inclined track.

First you must solve the previous equation for time:

𝑡 = √ 2𝒔

𝒂

If the object in our experiment was in freefall you would just need to substitute

the distance it was falling for s and substitute the acceleration due to Earth’s gravity for 𝒂, which is

g = 9.8 m/s2

In this experiment, however the object is not undergoing freefall, it is rolling down

an incline.

The acceleration of an object sliding, without friction down an incline is given by:

𝒂 = gSINθ

Where θ is the angle between the horizontal plane (the surface of your table)

and the inclined plane (the track), and g is the acceleration due to Earth’s

gravity.

When a solid sphere is rolling down an incline the acceleration is given by:

𝒂 = 0.71 gSINθ

The SIN (trigonometric sine) of an angle can be found by measuring the angle

with a protractor and using the SIN function on your calculator or simply by

dividing the length of the side opposite the angle (the height from which the

sphere starts) by the length of the hypotenuse of the right triangle (the length of

the track). Figure 9 shows the formula for deriving sines from triangles.

Note: Read the following section carefully.

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Figure 9

sin 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

Activity 4: Procedure

1. Set up the angle bar as a track. Measure the length of the track and the angle of

elevation between the track and the table.

2. Rearrange the kinematics equation to solve for time (second equation), and

substitute the value 0.71 g SINθ for 𝒂 (third equation). Use a distance of 80 cm for s.

𝒔 = 1

2 𝒂∆𝑡2

𝑡 = √ 2𝒔

𝒂

𝑡 = √ 2𝒔

(0.71𝐠 SINθ)

3. Release the steel sphere from the start point at the elevated end of the track and

measure the time it takes for the sphere to roll from position s = 0 to a final position s

= 80 cm.

4. Compare the measured value with the value predicted in Step 2. Calculate the

percent difference between these two numbers.

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5. Repeat Activity 4 with the acrylic sphere. What effect does the mass of the sphere

have on the acceleration of the object due to gravity?

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Activity 5: Demonstrate that a sphere rolling down the incline is

moving under constant acceleration

1. Collect the piece of foam board. Use a ruler and a pencil to draw lines across the

short dimension (width) of the board at 5 cm increments.

2. Collect rubber bands from the central materials kit. Wrap the rubber bands around

the width of the foam board so that the rubber bands line up with the pencil marks

you made at the 5 cm intervals. See Figure 10, left panel.

3. Use a book to prop up the foam board as an inclined plane at an angle from 5° to

10° from the horizontal.

4. Place the steel sphere at the top of the ramp and allow the sphere to roll down the

foam board.

5. Remove the rubber bands from the foam board.

6. On the reverse side of the foam board, use a pencil to mark a line across the short

dimension of the board 2 cm from the end. Label this line zero. Mark lines at the

distances listed in Table 5. Each measurement should be made from the zero line.

(see Figure 10).

Note: The sound as the steel sphere crosses the rubber bands will increase in

frequency as the steel sphere rolls down the ramp, indicating that the sphere is

accelerating. As the sphere continues to roll down the incline, it takes less time to

travel the same distance.

If the steel sphere is moving under a constant acceleration, then the displacement

of the sphere from the initial position, if the sphere is released from rest, is given by:

𝒔 = 1

2 𝒂∆𝑡2

The displacement at each time t should be proportional to 𝑡2

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Table 2

Displacement (cm)

1

4

9

16

25

36

49

64

81

7. Place rubber bands on the foam board, covering the pencil lines you just made.

8. Set the foam board up at the same angle as the previous trial.

9. Roll the steel sphere down the foam board.

Note: The sounds made as the sphere crosses the rubber bands on the foam board

in the second trial should be at equal intervals. The sphere is traveling a greater

distance each time it crosses a rubber band, but the time interval remains constant

meaning the sphere is moving with a constant acceleration.

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Figure 10

Note: For more information on the Trigonometry, Kinematics Equations, and

Rotational Motion exercises, visit the Carolina Biological Supply website at the

following links:

Basic Right Triangle Trigonometry

Derivation of the Kinematics Equations

The Ring and Disc Demonstration

http://www.carolina.com/teacher-resources/Interactive/basic-right-triangle-trigonometry/tr31008.tr?question=right%20triangle
http://www.carolina.com/teacher-resources/Interactive/derivation-of-the-kinematics-equation/tr32615.tr?question=kinematics
http://www.carolina.com/teacher-resources/Document/ring-and-disc-demonstration/tr32424.tr
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