ANALYSIS OF STUDENT CONCEPT KNOWLEDGE IN

KINEMATICS

Andreas Lichtenberger, Andreas Vaterlaus and Clemens Wagner

ETH, Department of Physics, Zurich, Switzerland

Abstract: We have developed a diagnostic test in kinematics to investigate the student

concept knowledge at the high school level. The 56 multiple-choice test items are

based on seven basic kinematics concepts we have identified. We perform an

exploratory factor analysis on a data set collected from 56 students at two Swiss high

schools addressing the following issues: What factors do the data reveal? What are the

consequences of the factor analysis on the teaching of kinematics? How can this test

be included in a kinematics course?

We show that there are two basic mathematical concepts that are crucial for the

understanding of kinematics: the concept of rate and the concept of vector (including

the direction and the addition of vectors). Furthermore the investigation of items with

different representations of motion (i.e. stroboscopic pictures, table of values and

diagrams) reveals that the students use different concepts for the different

representations. In particular, there seems to be no direct transfer between the

picture/table representation and the diagram representation.

Finally, we show how the test can be used as a diagnostic tool in a formative way

providing useful feedback for the students and for the teacher. By means of a latent

class analysis we identify four classes of students with different kinematics concepts

profiles. Such a classification may be helpful for teachers in order to prepare adjusted

learning material.

Keywords: Kinematics, Concept Knowledge, Diagnostic Test, Exploratory Factor

Analysis

INTRODUCTION

In the last two decades investigations in physics teaching at the high school and

undergraduate level have shown that a majority of science students have difficulties to

understand physics concepts (Hake, 1998; Halloun & Hestenes, 1985). Students often

attend classes with solid initial misconceptions. Conventional physics instruction

produces only little changes in their conceptual knowledge. The students may know

how to use formulas and calculate certain numerical problems but they still fail to

comprehend the physics concepts. The mentioned studies indicate that instruction can

only be effective if it takes into account the student preconceptions. The proper

concepts have to be learned but also the misconceptions have to be unlearned

(Wagner & Vaterlaus, 2011). This requires the diagnosis of student concepts and

misconceptions.

We have designed a diagnostic test with the purpose of identifying the student

concepts and misconceptions in kinematics at the high school level. The test is based

on the following list of kinematics concepts:

C1: Velocity as rate

C2: Velocity as vector in one dimension (i.e. direction of the velocity)

C3: Vector addition of velocities in two dimensions

C4: Displacement as area under the curve in a velocity-time-graph

C5: Acceleration as rate

C6: Acceleration as vector in one dimension (i.e. direction of the acceleration)

C7: Change in velocity as area under the curve in an acceleration-time-graph

The list of concepts has been verified by experts and is in good agreement with the

concepts identified in other studies (e.g. Hestenes, Wells & Swackhamer, 1992).

The development of a new kinematics test has been necessary, since so far there exists

no test that allows measuring the student concept knowledge for each concept

separately. The FCI (Hestenes, Wells & Swackhamer, 1992) and the MBT (Hestenes

& Wells, 1992) are mainly used as tests to evaluate the overall dynamics concept

knowledge. They actually both contain items that correspond to the concepts

mentioned above. However, the number of these items is too small to analyze each

concept separately. The Motion Conceptual Evaluation (Thornton & Sokoloff, 1998)

and the Test of Understanding Graphs in Kinematics (Beichner, 1993) on the other

hand are rather based on task-related objectives than on concepts. The items can

therefore not clearly be linked to the concepts listed above.

We have analyzed student responses to our kinematics test addressing the following

questions: Is the test a valid instrument to determine student concept knowledge about

kinematics? Do the students answer coherently referring to the suggested concepts?

What are the consequences of the test results on teaching?

In order to treat the first two issues we carry out an exploratory factor analysis similar

to the factor analysis of the FCI data done by Scott and Schumayer (2012). Factor

analysis is a standard technique in the statistical analysis of educational data sets and

is described in detail in many pieces of literature (e.g. Merrifield, 1974, Bühner,

2011). The goal of a factor analysis is to explain the correlations among the items in

terms of only a few fundamental entities called factors or latent traits. A latent trait is

interpreted as a characteristic property of the students and made visible while

attempting to answer the items. The degree to which a student possesses a particular

trait determines the likelihood to answer a particular item correctly. Thus the items are

the manifested indicators of the latent factors. Scott and Schumayer (2012) point out

that it is important to distinguish between "factors" and "concepts". In our context the

concepts are constructs defined by experts while the factors represent the coherence of

the student thinking. The interesting issue is whether the association of items seen by

an expert agrees with the association of questions seen by students.

Referring to the third issue we suggest how the test can be applied at school in a

formative way. We show how by means of a latent class analysis different groups of

students with a similar profile of concept knowledge can be found and how a

characterization of these groups can help the teacher to prepare individual material for

every group.

The following section describes the methods including the test instrument, the

collection of data and the exploratory factor analysis. Thereafter the results of the

factor analysis are presented and interpreted. The last two sections are devoted to the

application of the instrument at school and to a final discussion of the results.

METHODS

Test Instrument

The kinematics diagnostic test is designed for high school students at level K-10. The

test items are based on the list of concepts presented in the previous section. To every

concept there is also a set of corresponding misconceptions. The misconceptions have

been verified by asking the students open questions and by analyzing their answers.

Furthermore they have been confirmed by experts.

The test consists of 56 multiple-choice items on kinematics, each item containing one

right answer and three to four distractors. Every distractor has been chosen in a way

that it can be assigned to a single misconception. This is different from the other

kinematics tests mentioned before. Thus the test not only uncovers student concepts

but also student misconceptions. The items can be furthermore divided into three

levels of abstraction:

Level A: Items with images (e.g. stroboscopic pictures)

Level B: Items with tables of values

Level C: Items with diagrams

For all levels of abstraction a representative test item is presented in the appendix.

Prior to the explorative factor analysis we empirically verified the data set. In a first

step we sorted out all items with a difficulty above 0.85 or below 0.15 because items

with such high or low difficulties do not serve as good discriminators. As a second

step we determined the internal consistency calculating Cronbach's alpha for each

concept separately. We reviewed every item that did not contribute to the internal

consistency with respect to its content. Items that were considered capable of being

misunderstood or referring to multiple concepts were dropped.

Table 1

Distribution of the items. The stars mark items that refer to two concepts. Concepts 4

and 7 are completely excluded from further analysis.

Items Level A Level B Level C Total

Concept 1 2 4 5 51 53 35 36 40 46* 47* 10

Concept 2 6 7 28 42 46* 47* 6

Concept 3 8 9 10 3

[Concept 4] [16 29 37 43] [4]

Concept 5 11 12 54 38 39 48* 49* 7

Concept 6 13 14 44 48* 49* 5

[Concept 7] [18 45] [2]

Total 12 3 12 27

Through the whole verification process the data set was finally reduced from 56 to 27

items. The Table 1 shows the distribution of the remaining test items according to the

concepts and levels of abstractions. The numbers indicate the item numbers in the

test. The stars mark items that refer to two concepts. As the Cronbach's alpha

coefficients for the concepts 4 and 7 are below 0.30 these concepts are excluded from

further analysis. The Cronbach's alphas for the other concepts are between 0.60 and

0.80, the mean inter-item-correlations are between 0.21 and 0.41.

Collection of data

We collected the data from 56 students from classes of two teachers at two Swiss high

schools in autumn 2012. The average age of the participants was 16 years with a

standard deviation of 1 year and a range from 14 to 18 years. 30 participants were

female, 26 were male. About half of the students were majoring in economics, the

others in science and languages. Independent of their major subject all of the students

attended a similar basic kinematics course over about six weeks. The test was

presented online at the end of the instruction. The order of items was the same for all

students. They were required to complete the survey and no item could be skipped.

The time to answer the items as well as the time to complete the test was recorded

individually. The average overall time for completing the 56 items was (46 ± 8) min.

Exploratory Factor Analysis

For the analysis we used the program SPSS (2010). The first step of an exploratory

factor analysis is the construction of the correlation matrix between the set of items

that are investigated. We used the standard Pearson correlation function to calculate

the matrix. As a next step we chose the maximum-likelihood method for the reduction

of the correlation matrix. It is one of the standard methods and usually provides

similar results to the principal axis analysis. Moreover this method is mostly used also

in confirmatory factor analysis (Bühner, 2011). Before reducing the matrix, we

determined the optimal number of factors. This may be achieved in several ways. We

used the scree plot technique (Cattell, 1966), where the eigenvalues of the correlation

matrix are plotted in decreasing order. Scanning the graph from left to right one can

look for a knee. Then all the factors on the left of the knee are counted, while those

factors, which fall in the "scree" of the graph, are discarded (Bortz, 1999). An

example is given in Figure 1. The graph illustrates the eigenvalue of each successive

factor in the explanation of the data set. The full lines are guides to the eye. The

graphic suggests a three-factor model. Of course this method is somehow subjective,

as the "knee" is not defined accurately. Therefore it is important to prove if the

number of factors can be derived also from a theoretical model (Bühner, 2011).

The last step in factor analysis was to perform a rotation of the factor axes to see if

there was another set of eigenvectors, which is more amenable to interpretation. There

are two possibilities in rotating the axes. If we can assume that the factors are

uncorrelated, we require the resulting eigenvectors to be orthogonal. Alternatively, if

we do not know if the factors are correlated we allow the rotation to produce a set of

non-orthogonal eigenvectors. The latter option provides us with information about the

relationship between the factors. We used this option choosing the prevalent Promax-

method.

In order to carry out an exploratory factor analysis it is a standard rule of thumb to use

at least 10 times as many respondents as there are items in the test (Everitt, 1975). As

our sample size (N = 56) is relatively small concerning the number of items (n = 27)

we decided to make the analysis step by step. We first conducted the analysis for the

data of different abstraction levels A, B and C separately. Moreover we left out the

items 46-49 which refer to two concepts. This way the number of items was reduced

to 12, 3 and 8 for level A, B and C, respectively. Afterwards we checked if the results

for the different levels were compatible. In order to check if the set of items was

applicable to an exploratory factor analysis we calculated the Kaiser-Meyer-Olkin-

coefficients (Cureton & D’Agostino, 1983). The standard rule is that the KMO-

coefficient should be at least above 0.60, for good results yet above 0.80. Our values

ranged from 0.65 to 0.77.

Figure 1. Scree Plot. The eigenvalues of the Pearson correlation matrix are depicted

in decreasing order. The knee is between the factors three and four. This suggests a

three-factor model.

RESULTS AND INTERPRETATION

Level A items

For the level A items the scree plot (see Fig. 1) suggests that three factors determine

student responses. These three factors account for 47 % of the variance in the data.

The data analysis shows that all items (except item 12) can be clearly assigned to one

of the underlying factors. The loadings of these items onto the factors are between .33

and 1.00. We note that the items 2, 4, 5 and 11, which are grouped into factor 1, refer

to the rate concepts C1 and C5. Furthermore, the items 6, 7, 13, 14, which are grouped

into factor 2 refer to the vector concepts C2 and C6. Finally the items 8, 9, and 10

corresponding to concept C3 load on a separate factor 3. The correlation coefficients

between the factors are in the present non-orthogonal three-factor model 0.19

(between factors 1 and 2), 0.31 (between 1 and 3) and 0.38 (between 2 and 3).

The factor structure shows that there is a tendency for a student to give a correct

answer to one of the "rate items" (2, 4, 5, 11) given that this student has answered

another rate item correctly. The same holds for the "vector items" (6, 7, 13, 14) and

for the "vector addition items" (8, 9, 10). We may draw the conclusion that the

association of items seen by the students is in accordance with the association of

questions seen by experts. Moreover the actual contents velocity and acceleration

seem to play only a limited role. Much more relevant for answering the items

correctly is the understanding of the mathematical concepts of rate and vector

(including direction and addition). It is therefore tempting to interpret the underlying

factors as "rate concept", "direction concept" and "vector addition concept". The three

factors are only marginally correlated meaning that we have three almost independent

factors. The fact that the correlation is the strongest between the factors 2 and 3 is in

line with our interpretation. These factors both refer to a vector concept whereas

factor 1 refers to a rate concept.

Level B items

Level B (tables of values) only contains three items. A factor analysis indicates that a

single factor may be taken as underlying student responses. The factor explains 50 %

of the variation in the data. The loadings of the items 51, 53 and 54 on the factor are

0.85, 0.69 and 0.56. We note that all the items have high loadings on the factor.

However, the loading of item 54 is the lowest. The items 51, 53 and 54 are related to

the rate concepts C1 and C5 (see Tab. 1).

Again there seems to be an underlying "rate concept" which can explain a notable part

of the correlation of the items 51, 53, 54. The fact that item 54 has a lower loading

may be due to its different content. While the items 51 and 53 are about velocity, item

54 polls student understanding of acceleration.

Level C items

Considering the scree-plot, we used a two-factor model for the data from the level C

items. The two factors account for 47 % of the variance in the data. All items can be

clearly assigned to one of the underlying factors. The factor loadings range from 0.29

to 0.96. The correlation coefficient between the factors is 0.301.

We find again that the items corresponding to the rate concepts C1 and C5 group into

one factor whereas the items linked to the direction concepts C2 and C6 group into

another one. As for solving the items with stroboscopic pictures (level A) also for

solving the diagram items (level C) there seem to be two underlying factors that may

be interpreted as a “rate concept” on one side and a “direction concept” on the other

side. Of course it is not clear if the factors found in the two different levels A and C

are actually the same. But again the understanding of the two basic mathematical

concepts of rate and direction seems to be crucial for the interpretation of diagrams in

kinematics. The correlation coefficient between the factors is again small indicating

that the two factors are mostly independent of each other.

Overall result

The interesting issue is whether the "rate factors” and the “direction factors” found at

different abstraction levels are correlated: Are these two factors universal for solving

problems in kinematics? In order to investigate this issue we carried out a factor

analysis including all items, which loaded on these two factors at levels A, B and C.

The result of this analysis is shown in Table 2. Four factors were detected explaining

50.0 % of the total variance in the data set. It is common practice to accept loadings

above 0.3 as indicating a relevant correlation between a particular item and the

underlying factor (Kline, 1994). Therefore and for better clarity, absolute values

below 0.3 are either hidden or put in brackets, if they are important for interpretation.

The first factor groups together the items from level A and B corresponding to the rate

concepts C1 and C5. With exception of item 12, which also loads on factor 3, the

loadings are all between 0.59 and .99 meaning that these items have a high correlation

with the underlying factor. The second factor mainly groups the items from level A

and B, which refer to the direction concepts C2 and C5. However, item 7 loads on all

the factors and cannot be assigned clearly to one factor. The factors 3 and 4 group the

items of level C. Again there is a tendency that the items corresponding to the rate

concepts contribute to one factor whereas the items referring to the direction concepts

load on the other factor. The highest factor correlation is between the factors 2 and 3

with a value of 0.42. The other correlations are below 0.3.

Table 2

Factor loadings for all factor 1 and factor 2 items of the levels A-C.

Level Item Factor Corresponding Concept

1 2 3 4

A 2 .78

A 4 .68 C1: Velocity as rate

A 5 .99

A 11 .60 C5: Acceleration as rate

A 12 [.27] [-.27] .35 [-.21]

B 51 .70 C1: Velocity as rate

B 53 .71

B 54 .59 C5: Acceleration as rate

A 6 .37 C2: Velocity as vector

A 7 [.08] [.18] .35 .33

A 13 .69 C6: Acceleration as vector

A 14 .69

C 35 [.15] .40 [.24] [-.16]

C 36 .67 .37 C1: Velocity as rate

C 40 [.13] [-.03] [.28] [.13]

C 38 .72 C5: Acceleration as rate

C 39 .92

C 28 .30 C2: Velocity as vector

C 42 .98

C 44 .47 C6: Acceleration as vector

The main observation is that we have different factors for level A/B and level C items.

Obviously, from the students point of view the interpretation of diagrams differs from

the interpretation of stroboscopic pictures and tables. There is no direct transfer

between these two representations of motion. Therefore instead of having two

universal rate and direction factors we have to distinguish between the levels of

abstraction or, in other words, between the different representations. Overall there

seem to be five different underlying factors that are determining the correct answering

of the items. We suggest interpreting the factors as follows:

Factor 1: “Rate concept” for representations with images and tables

Factor 2: “Direction concept” for representations with images and tables

Factor 3: “Rate concept” for representations with diagrams

Factor 4: “Direction concept” for representations with diagrams

Factor 5: “Vector addition concept” for representations with images

There are some details in the results that need to be discussed. First item 12 does not

mainly load on factor 1. There is no indication that the item differs from the other

factor 1 items as regards form and content. A possible reason is the high difficulty of

.80. As discussed before high difficulties usually lead to smaller correlations, in

particular when the sample size is rather small. Also item 7 does not fit well into our

suggested 5-factor-model. Obviously the integration of the level C items into the

factor analysis slightly changes the factor axes such that the loading of item 7 on the

factor 2 is lowered. There is no obvious reason why item 7 loads on the factors linked

to the diagrams. We have to recall that the sample is actually to small for the number

of items included in the present factor analysis such that the values have to be

interpreted with caution. Finally on level C we have the items 35 and 36, which do not

only load on the rate factor anymore but also on the direction factor. This fact is

actually due to item 40. After removing that item from the analysis we discovered an

increase of the loadings of items 35 and 36 on the rate factor. This shows again that

the factor analysis is very sensitive to small changes when the number of items is big

compared to the sample size. The loadings of the items 35, 36 and 40 on both the

factors 2 and 3 are also the cause for the noted correlation between the factors 2 and 3.

There is no obvious reason for this correlation from a theoretical point of view.

At last we investigated how the items 46 – 49, which can be linked to both the rate

concept and the direction concept, fit into our 5-factor-model. All of these items

contain a given kinematics graph (e.g a velocity-time diagram). The student then has

to select another corresponding diagram (e.g. a position-time diagram). We integrated

the items one by one to check which factor they load on while the factor axes are not

changed too much. We found that all these items load on both the factors 3 and 4 with

values above 0.3. This is an important finding as it shows that also the answering to

items that are referring to more than one concept can be explained within our 5-factor-

model. There is no indication that new factors emerge for more complex problems.

APPLICATION

We suggest integrating the present test in the basic kinematics course in a formative

way. The test provides a detailed feedback for the students as well as for the teacher.

For every student, two diagrams can be prepared, one illustrating the percentages of

items solved correctly for each of the seven concepts and the other showing which

misconceptions are still present. The teacher gets feedback about the overall

performance of the class. Furthermore by means of a latent class analysis (LCA) the

teacher can find groups of students with similar concept profiles (Collins & Lanza,

2010). This allows the teacher to prepare customized materials for the groups such

that the students can work on their individual deficits having the chance to catch up.

For better illustration we performed a LCA with help of the program MPlus (2011).

We included the data of the 27 items shown in Table 1. In order to determine the

optimal number of classes we used a technique similar to the one used for the factors.

Instead of plotting the eigenvalues, we plotted the loglikelihood against the number of

classes. By locating the knee in the graph we found four different classes that can be

assigned to four groups of students. The characteristics of the four groups are shown

in Figure 2. The mean score is defined as the group average of the fraction of

correctly solved items corresponding to the particular concept. Even if we did not

include the items referring to the concepts C4 and C7 in the LCA, we plotted the

mean scores for completeness. The four groups can be characterized as follows:

Class 1: “All-rounders”. These students understand all concepts sufficiently.

Class 2: “1D-students”. These students solve problems in one dimension often

correctly, but they fail at the two-dimensional addition of velocity vectors.

Class 3: “Non-Vectorians”. These students seem to have a good understanding

of the rate concepts but they have difficulties with the vector concepts

(direction and addition).

Class 4: “Conceptless”. These students are not able to apply a concept in

different situations properly.

Figure 2. Characteristics of the classes found with LCA.

We suggest that the diagnostic test is followed by a reflective lesson where the

students are given time to work on their deficits. The classification simplifies the

preparation of individualized learning material. The teacher directly has an overview

of the characteristic groups of students and he can provide adjusted learning material.

For example, the teacher can prepare material about the addition of vectors for all

students who belong to class 2.

Of course, these groups are only exemplary. Further studies are needed to investigate

if these characteristics are typically found in kinematics classes at Swiss high schools.

DISCUSSION

We have found that there are two basic mathematical concepts that are crucial for the

understanding of kinematics: the concept of rate and the concept of vector (including

direction and addition). The context and the content seem to play only a minor role. If

a student understands the concept of rate he is able to answer correctly to questions

about velocity and acceleration in different contexts. The same holds for the vector

concept. This result has direct implications for the instruction. It suggests that in

kinematics courses the focus should be first on the learning of the mathematical

concepts. Transferring the mathematical concepts to physical contents and applying

them in different contexts is suggested to be easier for students than learning physical

concepts without a mathematical fundament. These findings are somewhat in line

with the results of Christensen and Thompson (2012) who investigated the graphical

representations of slope and derivative among third-semester students. In the

conclusion they stated, that “some of their demonstrated difficulties [in physics] seem

to have origins in the understanding of the math concepts themselves”. Moreover also

Bassok and Holyoak (1989) found similar results analyzing the interdomain transfer

between isomorphic topics in physics and algebra. Students who had learned

arithmetic progressions were very likely to spontaneously recognize the application of

the algebraic methods in kinematics. In contrast, students who had learned the physics

topic first almost never exhibited any detectable transfer to the isomorphic algebra

problems. Finally, it has to be mentioned, that even if the understanding of the

mathematical concepts seems to be a requirement for understanding kinematics, it

does not guarantee success (Planinic, Ivanjek and Sussac, 2013).

Another interesting finding is that the expert associations of items corresponding to

the concepts C4 and C7 could not be found in the student answers. These items

involve the evaluation of areas under the curve. Obviously most of the student did not

have proper area concepts. Instead of that, interviews showed that students often

argued with a concept of average. For example when they were asked to interpret the

velocity-time-diagram of an object regarding to its covered distance, they often did

not consider the area under the curve but tried to estimate the mean velocity. From a

mathematical point of view, finding the mean value is equivalent to determining the

area under the curve and dividing by the interval size. Still, the interviews indicated

that the use of an average concept is accompanied by different misconceptions than

the use of the area concept. All in all the items corresponding to concept C7 were the

most difficult of the test. This can be seen in Figure 2. These results are in line with

the findings of Planinic, Ivanjek and Susac (2013). They also found that the slope

concept (which we call the rate concept) could be easily transferred from

mathematical to physical contexts. However, this is not the case for the area under the

graph concept. The transfer of this concept from mathematics to physics was found to

be much more difficult for the students. A possible reason could be “the fact that

during the teaching of kinematics the interpretation of the slope is usually emphasized

much more than the interpretation of the area under the graph”.

As the kinematics test used in this study contains 27 items, a minimum number of 270

students is needed to produce a reliable result by means of a factor analysis. As we do

not meet this requirement (N = 56), the present results are preliminary. Still, the fact

that the association of items given by the assignment to the concepts by experts could

be clearly found in the student answers is very promising. Furthermore most of the

results in this study confirm results from other studies. This gives rise to hope that the

results will be corroborated in a following study with a bigger sample size.

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APPENDIX

Example 1: Item 14 (Level A, concept C6: acceleration as vector)

A helicopter is approaching

for a landing. It moves

vertically downwards and

reduces its velocity.

Which of the following statements describes the acceleration of the helicopter best?

1. The acceleration is zero.

2. The acceleration points downwards.

3. The acceleration points upwards.

4. The direction of the acceleration is not defined

5. The acceleration has no direction.

Example 2: Item 51 (Level B, concept C1: velocity as rate)

Two bodies are moving on a straight line. The positions of the bodies at successive 0.2-

second time intervals are represented in the table below.

Time in s 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Body 1: Position in m 0.2 0.4 0.7 1.1 1.6 2.2 2.9 3.7

Body 2: Position in m 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8

Do the bodies ever have the same speed?

1. No.

2. Yes, at the instants t = 0.2 s and t = 0.8 s.

3. Yes, at the instants t = 0.2 s.

4. Yes, at the instants t = 0.8 s.

5. Yes, at some time between t = 0.4 s and t = 0.6 s.

Example 3: Item 28 (Level C, concept C2: velocity as vector)

The following represents a position-time graph (x-t-diagram) for an object.

Niveau:(4( Quelle:(vgl.(Beichner(3(

s"t((((

((((((

(((((

(

a. Der&Körper&bewegt&sich&eine&schiefe&Ebene&hoch.&b. Der&Körper&bewegt&sich&ausschliesslich&rückwärts.&c. Der&Körper&bewegt&sich&zuerst&rückwärts,&dann&vorwärts.&d. Der&Körper&bewegt&sich&ausschliesslich&vorwärts.&

Niveau:(4( Quelle:(vgl.(Beichner(3(

s"t((

((

((((((

((((

(

(

a. Der&Körper&bewegt&sich&ausschliesslich&vorwärts.&b. Der&Körper&bewegt&sich&ausschliesslich&rückwärts.&c. Der&Körper&bewegt&sich&zuerst&vorwärts,&dann&rückwärts.&d. Der&Körper&bewegt&sich&eine&schiefe&Ebene&hinunter.&

+(

(((((

(–(

+(

(((((

(–(

x&

t&

Which of these describes the motion best?

1. The object always moves forward.

2. The object always moves backward.

3. The object moves forward at first. Then it moves backward.

4. The object moves down an inclined plane.

Which of the following statements describes the