International Journal of Psychology, 2015
Vol. 50, No. 4, 256–264, DOI: 10.1002/ijop.12095
Developmental changes in children’s understanding of
horizontal projectile motion
Yi Mou1,2 , Liqi Zhu1 , and Zhe Chen3
1 Key
Lab of Behavioral Science, Institute of Psychology, Chinese Academy of Sciences,
Beijing, China
2 Department of Psychological Sciences, University of Missouri-Columbia, Columbia, MO, USA
3 Department of Human and Community Development, University of California-Davis, Davis,
CA, USA
T
his study investigated 5- to 13-year-old children’s performance in solving horizontal projectile motion problems, in
which they predicted the trajectory of a carried object released from a carrier in three different contexts. The results
revealed that 5- and 8-year-olds’ trajectory predictions were easily distracted by salient contextual features (e.g. the relative
spatial locations between objects), whereas a proportion of 11- and 13-year-olds’ performance suggested the engagement
of the impetus concept in trajectory prediction. The impetus concept is a typical misconception of inertial motion that
assumes that motion is caused by force. Children’s performance across ages suggested that their naïve knowledge of
projectile motion was neither well-developed and coherent nor completely fragmented. Instead, this study presented the
dynamic process in which children with age gradually overcame the influences of contextual features and consistently
used the impetus concept across motion problems.
Keywords: Naïve physics; Impetus theories; Misconception; Horizontal projectile motion.
What do children and adults untrained in physics know
about the physical world they are living in? Research on
this issue demonstrates that one understands some basic
physical concepts (e.g. the principles of spatiotemporal
continuity, object solidity and cohesion) from early life
and spontaneously uses these naïve concepts to predict
and interpret everyday physical phenomena (Baillargeon,
2008; Hood, 1995; Ioannides & Vosniadou, 2001; Kim &
Spelke, 1999; Spelke, 1994; Wilkening & Huber, 2002).
While children and adults can correctly predict and interpret some physical events, they also have misconceptions
for seemingly simple physical phenomena, such as force
and inertial motion (Clement, 1982; Kaiser, Proffitt, &
McCloskey, 1985; Krist, Fieberg, & Wilkening, 1993;
McCloskey, 1983a; Mou & Zhu, 2006).
Individuals’ naïve understanding of force and inertial motion is typically investigated in horizontal projectile motion problems, in which participants predict
the trajectory, speed and/or landing location of a moving object in horizontal projectile motion (Kaiser et al.,
1985; Kim & Spelke, 1999; Krist, 2000; McCloskey,
1983a, 1983b; McCloskey, Washburn, & Felch, 1983).
McCloskey’s seminal research (e.g. McCloskey, 1983a,
1983b; McCloskey et al., 1983) found that some college students erroneously predicted that a carried object
dropped from a moving carrier would fall straight down
(the straight-down belief in the carried object context),
instead of along the actual forward-arcing parabola. These
participants reported that the falling object received no
force to maintain its forward motion and thus fell straight
down due to gravity (McCloskey, 1983b). In addition,
some participants correctly predicted forward movement,
but they still attributed it to the force the carried object
obtained from the carrier (McCloskey, 1983b). Therefore,
even though participants predicted different trajectories
(straight down or forward), they showed the same misconception of motion: motion must require force.
This misconception is substantially against the Newtonian inertia principle, but resembles the impetus
theory, the medieval perspective about force and motion
(Clement, 1982; McCloskey, 1983a). The impetus theory
claims that it is the impetus, which is imparted to an
Correspondence should be addressed to Liqi Zhu, Key Lab of Behavioral Science, Institute of Psychology, Chinese Academy of Sciences, Beijing
100101, China. (E-mail: [email protected]).
This research was supported by Grant KJZD-EW-L04 from the Chinese Academy of Sciences.
© 2014 International Union of Psychological Science
UNDERSTANDING OF PROJECTILE MOTION
object by external force, that maintains the motion of the
object. In addition, the impetus gradually dissipates during motion and the exhaust of impetus causes the cease of
motion. Because adults’ impetus misconception has been
commonly observed in different motion problems with
different response measurements (e.g. paper-and-pencil,
dynamic animation and actual action tests) (Eckstein &
Shemesh, 1993; Kaiser, Proffitt, Whelan, & Hecht, 1992;
McCloskey, 1983a, 1983b; McCloskey et al., 1983), it is
proposed that people may have a general and coherent
impetus-based mental model and they consistently use
the model to predict object motion in different motion
problems (McCloskey, 1983a, 1983b; McCloskey et al.,
1983).
In contrast, some other studies show that individuals’ trajectory predictions are not consistent across
different motion problems and are easily influenced by
problem contexts (Cooke & Breedin, 1994; DiSessa,
1988, 1993). For example, while participants show the
straight-down belief when predicting the trajectory of
an object dropped from a horizontally moving carrier or
from a swinging pendulum, they correctly predict that
a ball rolling off a table will continue moving forward,
even though these problems are isomorphic in physical
principles (Cooke & Breedin, 1994; Kaiser et al., 1985).
In addition, participants’ explanations for their predictions suggest that they do not consistently engage the
impetus concept across problems. Instead, they primarily
use contextual features presented in the problems to make
trajectory predictions, which explains why participants’
responses vary across problems (Cooke & Breedin,
1994). In addition to different types of motion problems,
participants’ trajectory predictions are also influenced
by how the problems are presented and how the participants’ responses are measured. For example, participants
generally make better trajectory predictions in motion
problems presented in dynamic than in static presentations (Kaiser et al., 1992), and better when selecting than
producing trajectories (e.g. Cooke & Breedin, 1994).
In sum, the aforementioned findings suggest that individuals do not consistently engage the impetus concept
to make trajectory predictions across different motion
problems. Instead, individuals’ physical knowledge is
only piecemeal and their predictions and interpretations
on physical events are task-specific (Cooke & Breedin,
1994; DiSessa, 1988, 1993).
The debate of whether individuals engage the
impetus-based concept to predict projectile motion
has been complicated by developmental studies. On the
one hand, the straight-down belief is commonly seen
in children’s trajectory predictions for the horizontal
projectile problems (Howe, Taylor Tavares, & Devine,
2012; Kaiser et al., 1985; Krist, 2000), which suggests
the engagement of the impetus concept in children. For
instance, when seeing a ball roll off a table or being
dropped from a moving carrier, about half of children
© 2014 International Union of Psychological Science
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in both cases put a bowl right under the point where the
object started to fall to catch it (Kaiser et al., 1985). In
another study, when children were asked to drop an object
to hit a target on the ground while they were moving, they
tended to release the object when their hands were right
above the target (Krist, 2000). Paper-and-pencil tests also
showed that over half of second to fourth graders drew the
straight-down trajectory for a rolling ball regardless of
its initial velocity (Eckstein & Shemesh, 1993). Notably,
these studies also show that children’s straight-down
belief declines with age (cf. the carried object problem in
Kaiser et al., 1985). For example, while the majority of 6and 8-year-olds in Krist (2000) showed the straight-down
belief, over half of 12-year-olds predicted that the object
would continue moving forward when falling and so
they released the object earlier than the younger children.
However, due to the lack of children’s verbal explanations, it is unclear whether the older children attributed
the forward movement to the impetus, just as some adults
did (McCloskey, 1983a, 1983b).
On the other hand, children’s trajectory predictions,
like adults’, vary across projectile problems, suggesting
that their naïve physics knowledge of projectile motion
is fragmented. Children’s performance was not correlated
across 16 isomorphic projectile motion problems, suggesting that they did not consistently engage the impetus concept in these problems (e.g. Anderson, Tolmie,
Howe, Mayes, & Mackenzie, 1992). Their trajectory predictions differ even just between the rolling and the carried object problems: while children with age were more
likely to predict the rolling ball to continue moving forward, they stuck to the straight-down belief in the carried
object problem (Kaiser et al., 1985). Furthermore, children’s performance varies even with the physical properties of the falling object. They show the straight-down
belief for heavy and slow falling objects but not for light
and fast ones (Anderson et al., 1992).
In addition, children perform differently when making
predictions in real action or in conceptual judgement.
Children from 5 to 10 years of age showed corrected trajectory predictions when throwing a ball from different
heights to hit a target on the ground, but made erroneous
predictions when explicitly rating the throwing speed in a
scale, even though the rating test was administrated right
after the action test (Krist et al., 1993). The disassociation
between participants’ action and judgement performance
suggests that there are two distinct types of knowledge, implicit perceptual-motor and explicit conceptual
knowledge, in individuals’ naïve physics (Krist et al.,
1993; Wilkening & Huber, 2002). The perceptual-motor
knowledge may derive from abstract representations of
body–environment relations and be accessed only in
motor actions (e.g. throwing a ball) but not in verbal,
conceptual judgement (e.g. rating the throwing speed).
Although, implicit perceptual-motor and explicit conceptual knowledge can be quite different and even conflicting,
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they can also be reconciled in one’s trajectory predictions
in some cases (e.g. children who hold the straight-down
belief really tended to drop the object right above the target; Krist, 2000; Krist et al., 1993). Nevertheless, the distinction between implicit perceptual-motor and explicit
conceptual knowledge suggests there is not a monolithic
structure of individuals’ naïve physics knowledge.
Taken together, neither the coherent mental model nor
the piecemeal knowledge views can solely explain the
existing mixed findings. The two conflicting views can be
integrated in a compromised view. That is, children, as
well as adults untrained in physics, may have the abstract
impetus concept, but they use it in only some but not
all projectile problems (Krist et al., 1993). Children and
adults do not consistently use the impetus concept across
seemingly very different isomorphic motion problems,
because they fail to recognise the isomorphism in their
own naïve physics. For example, because it is difficult to
recognise that an object dropped from a moving carrier
is physically isomorphic to a swinging pendulum being
cut off at the nadir, people who use the impetus concept to predict the trajectory of the carried object may not
use the same concept to predict the trajectory of the pendulum. Instead, they only respond to particular contextual features involved in the pendulum problem. In contrast, when motion problems have just slight differences
in contextual features such that the isomorphism can be
recognised, people may show more consistent predictions
across these problems. Consistent with this hypothesis, a
study showed that adults’ trajectory predictions were different across different types of motion problems but consistent within the types of problems with slight contextual
differences (e.g. only the motion orientation differed in
these problems; Cooke & Breedin, 1994). Children, like
adults, make quite different predictions across different
types of projectile problems, but it is unclear whether they
can make relatively consistent predictions on problems
within the same type and with slightly different contextual features, and whether these predictions are based on
the impetus concept.
Another approach to reconcile the conflicting views
is to investigate the developmental changes of individuals’ trajectory predictions in projectile problems.
Developmental studies suggest that children have some
abstract physical concepts (e.g. inertia, gravity and object
occlusion) from early in life, but they initially use these
concepts to predict very limited physical events. With
the increase of age and experience, children gradually
engage these concepts to predict more physical events
(Baillargeon, 2008; Spelke, 1994). One example is
that infants from 3.5 months of age understand that a
tall object cannot be fully hidden by a shorter object
(Baillargeon & DeVos, 1991), but they only make successful predictions in a context where a tall object is
hidden behind a shorter object. As they age and gain
more physical experience, they understand that the tall
object cannot be fully contained or covered by a shorter
one (see Baillargeon, 2008 for review). As for children’s
understanding of projectile motion, it is also possible
that children initially engage the impetus concept only
in limited projectile problems. However, they may be
able to gradually engage it to predict and interpret more
projectile problems during development.
These analyses raise two interesting questions. First,
how do children make trajectory predictions within the
same type of projectile problems (e.g. the carried object
context) with different contextual features? In particular,
do they simply respond to contextual features presented in
problems due to their piecemeal naïve physics knowledge
and thus show quite different predictions across the problems, or make consistent predictions across these problems on the basis of the impetus concept? Second, how
do children’s predictions across problems change with
age? Are children’s trajectory predictions in the projectile
problems fragmented or consistent during development or
do they show changes with age?
To shed light on these issues, the present study tested
children from 5 to 13 years of age on three carried object
problems with slight differences in contextual features:
a parallel movement, a hovering and a typical carried
object problem. The parallel movement problem presented a carrier, with a carried object, horizontally moving in parallel to a target object that was always under
the carrier. Children were asked in which one of three
directions the carried object should be launched in— forward, downward or backward—to successfully hit the
moving target, and they were also encouraged to explain
their choices. The child with the typical impetus concept might choose launching the object forward because
the object needs forward impetus to maintain its forward
motion to catch the moving target. However, according
to Newtonian mechanics, launching downward is the correct prediction (when air resistance is ignored) because
the carried object keeps the same horizontal velocity during falling as the moving carrier and the target.
However, children’s downward-launching choice
might have nothing to do with Newtonian mechanics but
simply due to their response to the salient contextual feature, the spatial relationship between the carrier and the
target (i.e. launching the object directly towards the target
under the carrier). To examine this possibility, children
were tested in a hovering problem, in which the carrier
hovered in the air without movement and it could launch
the carried object only when the target was moving to the
location under the carrier. The spatial relationship-based
strategy would lead to the same downward-launching
prediction as in the parallel movement problem. To make
the correct forward prediction, motion variables such
as the distances among objects and the time the carried
object took to hit the target must be considered.
In addition to their predictions of the launching directions, children were also presented with a typical carried
© 2014 International Union of Psychological Science
UNDERSTANDING OF PROJECTILE MOTION
object problem, in which only the moving carrier and the
carried object were presented. Children were asked to
predict the trajectory the carried object took after being
dropped from the carrier by choosing a trajectory from
several alternatives, and they also needed to explain their
predictions. Each individual’s performance in the three
problems was considered jointly to examine whether they
made consistent predictions across problems. Because
children in the three problems were required to predict
trajectories and give verbal explanations, their explicit
conceptual knowledge, not implicit perceptual-motor
knowledge, was assessed. The present experimental
design was adopted because it allowed detailed examination of children’s understanding of different trajectories,
which was difficult to obtain from action tasks.
METHOD
Participants
Four groups of children participated in this experiment:
thirty-one 5-year-olds (15 males; M = 5 years 6 months),
thirty 8-year-olds (14 males; M = 8 years 0 months),
twenty-nine 11-year-olds (15 males; M = 11 years
0 months) and thirty 13-year-olds (15 males; M = 13 years
4 months). To investigate children’s naïve understanding
of projectile motion, no children older than 14 years of
age (eighth grade) were recruited because Newtonian
mechanics is taught from then. The 5-year-olds were
recruited from a university-based preschool, the 8- and
11-year-olds were from a public elementary school and
the 13-year-olds were from a public middle school. All
schools are located in the urban area of Beijing, China.
Materials
The experiment consisted of three motion problems presented in dynamical animations on a computer. The participants were asked to choose a launching direction for a
carried object in the parallel movement and the hovering
problem, and to predict the trajectory of the carried object
in the typical carried object problem.
Parallel movement problem
The display showed a toy helicopter carrying a rubber bomb and flying at a constant speed. A toy tank
on the ground was always right under the helicopter
and the two were moving at the same speed in parallel. After they “rolled off” the right edge of the screen,
they reappeared from the left side and continued the
left-to-right motion. Three arrows below the carried bomb
indicated three potential launching directions: backward,
downward and forward (Figure 1). The experimenter presented the dynamic display to the participants and asked
© 2014 International Union of Psychological Science
259
Figure 1. Schematic of the parallel movement and the hovering problems.
them which of the three directions the rubber bomb should
be launched to hit the moving tank. The experimenter
emphasised that there was no wind in this context. To
enhance the visual reality of animation, some background
features such as landscape and clouds were included in
presentation. These features did not distract or confuse
children given that their verbal reports (see Procedure)
showed that none of the children included these features
to predict launching directions.
Hovering problem
The display was the same as the parallel movement
problem except that the helicopter was motionlessly hovering in the air and could launch the bomb to hit the tank
only when the tank was moving to the location right under
the helicopter.
Typical carried object problem
This display showed that the helicopter with the
bomb attached moved in the air. Participants were asked
to choose a trajectory from six alternatives the falling
bomb might take when being dropped from the moving
helicopter. The six trajectories were summarised from
previous studies (e.g. Eckstein & Shemesh, 1993; Kaiser
et al., 1985; McCloskey, 1983a) and presented on one
card (29.7 × 21 cm2 ; Figure 2). Children were told that
there was no wind in this context. In the pilot test, each
forward trajectory had a mirror backward trajectory, but
they were rarely chosen. Therefore, only one backward
trajectory with the highest frequency of selection was
kept in the formal test. The participants were also told
to draw their own trajectories if they thought none of the
trajectories seemed right.
Procedure
Children participated individually in a quiet room
at school. All participants first received the parallel
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Figure 2. The six trajectories presented in the typical carried object
problem. From upper left to lower right: back-diagonal trajectory (T1),
straight-down trajectory (T2), inversed L-shaped trajectory (T3); diagonal trajectory (T4); modified inversed L-shaped trajectory (T5) and
parabolic trajectory (T6).
movement and the hovering problems and then the typical carried object problem. Thus, the participants did
not receive any visual cues from the trajectory sketches
before answering the parallel movement and the hovering
problems. The presentation order of the parallel movement and the hovering problems was counterbalanced
across participants. While presenting the three problems,
the experimenter gave the instructions to the participants.
After the participants gave their answers, the experimenter encouraged them to explain their responses by
asking: “Why did you choose this direction/trajectory?”
None of the children in the typical carried object problem
chose to draw their own trajectories, but 40.8% of them
spontaneously replicated the chosen trajectories when
explaining their choices.
RESULTS
The choices and verbal reports in the parallel
movement problem
In the parallel movement problem, 93.5% of 5-year-olds
chose the downward option, suggesting that they thought
when the helicopter and the tank were in parallel movement, a downward launch allowed the bomb to hit the
tank (Figure 3). With increasing age, the proportion of
the downward choice decreased and that of the forward
choice increased (3.2, 20, 48.3 and 63.3% of 5-, 8-, 11and 13-year-olds, respectively, chose forward choice).
To focus on the developmental changes for the forward
choice, this option was compared with the downward
and the backward options in combination, and the result
showed the significant increase of the forward choice
with age, χ2 (3, N = 120) = 30.11, p < .01, φ = .50. No
significant sex difference or presentation order difference
was found (ps > .05).
Participants’ verbal reports were classified into five
categories: (a) Spatial position variable only: Children
reported only the relative spatial positions of the helicopter and the tank (e.g. "The helicopter and the tank
are aligned."); (b) Speed variable only: Children made
the predictions only based on the speed of the helicopter,
the tank or both (e.g. "The two run the same fast"); (c)
Several variables integrated: Children reported motion
variables such as distance, speed and time in both vertical
and horizontal motion (e.g. "When the bomb falls straight
down, the tank has already gone, so you must throw it
forward."); (d) Irrelevant variables: Reporting irrelevant
variables for their predictions (e.g. "The bomb has a
pointed end.") and (e) Other: No explanations or answering, "I don’t know." If a child reported multiple strategies,
each strategy was counted in the corresponding category.
Two raters coded participants’ reports and the interjudge
agreement was .91.
Five-year-olds’ main strategy, spatial position variable only, 48.4%, χ2 (4, N = 37) = 15.4, p < .01, showed
that their downward choice was largely based on the
salient spatial relationship between the helicopter and the
tank, but not on the inertia principle (Figure 4). This strategy, however, was gradually abandoned as children aged,
and the several variables integrated strategy was increasingly more reported by 11- and 13-year-olds, 44.8%,
χ2 (4, N = 30) = 15.33; 56.7%, χ2 (4, N = 32) = 26.13,
ps < .01, suggesting that the older children considered
the motion variables such as distance, speed and time
necessary for prediction. In addition, the speed variable
only strategy was favoured by 56.7% of 8-year-olds, χ2 (4,
N = 34) = 21.88, p < .01.
The choices and verbal reports in the hovering
problem
There were 83.8% of 5-year-olds choosing the downward
option in the hovering problem, and the spatial position variable only was their main strategy, 71%, χ2 (4,
N = 35) = 41.43, p < .001. These results again showed
that 5-year-olds focused on the salient spatial relationship
between objects. The majority of 8-, 11- and 13-year-olds
(63.3, 79.3 and 83.3%) correctly chose the forward option
(Figure 3), and the proportions significantly increased
with age, χ2 (3, N = 120) = 36.31, p < .01, φ = .55.
Correspondingly, the several variables integrated strategy was the main strategy reported by the three age
groups, 56.67%, χ2 (4, N = 32) = 22.68; 75.8%, χ2 (4,
N = 29) = 55.33; 80%, χ2 (4, N = 31) = 64.32, ps < .001.
The choices and verbal reports in the typical
carried object problem
Trajectory 2 (T2, the straight-down trajectory; Table 1)
was the dominant choice in all ages, χ2 (5, N = 31) = 79;
© 2014 International Union of Psychological Science
UNDERSTANDING OF PROJECTILE MOTION
261
Figure 3. The participants’ choices of the three launching directions in the parallel movement problem (a) and the hovering problem (b).
Figure 4. The participants’ strategies in the parallel movement problem (a) and the hovering problem (b).
χ2 (5, N = 30) = 54.4; χ2 (5, N = 29) = 27.9; χ2 (5,
N = 30) = 29.5, ps < .001. However, the proportion
of choosing T2 decreased from 77.4% in 5-year-olds
to 50% in 13-year-olds. The backward trajectory was
accepted by very few children in the four groups, and
thus T1 and T2 were combined as a non-forward category. Because of the small sample size in each of T3
to T6 options, these options were combined to form a
new forward category. Comparing the non-forward and
forward categories showed that more children chose
the forward trajectories with age, χ2 (3, N = 120) = 8.45,
p < .04, φ = .27. This increasing proportion was mainly
due to more 11- and 13-year-olds choosing T4 (the
diagonal trajectory) and T6 (the parabolic trajectory).
Very few children in the four age groups chose T3 (the
inverse-L shaped trajectory) or T5 (the modified inverse-L
shaped trajectory), suggesting that children did not think
the carried object continued moving horizontally after
release.
Children generally had difficulties in explaining their
trajectory predictions, and only 13-year-olds’ reports
were available for analyses. Seven of fifteen 13-year-olds
who chose T2 explicitly mentioned impetus (or force,
energy, power, etc.) in their reports (e.g. "no force is on
the bomb any more after it is dropped."). Among the 15
TABLE 1
The number of children choosing each of the six trajectories in the typical carried object problem
Age
5-year-olds
8-year-olds
11-year-olds
13-year-olds
T1
T2
T3
T4
T5
T6
1 (3)
2 (7)
1 (4)
0 (0)
24 (77)
20 (67)
15 (52)
15 (50)
0 (0)
1 (3)
4 (14)
0 (0)
2 (7)
3 (10)
4 (14)
8 (27)
3 (10)
2 (7)
1 (4)
1 (3)
1 (3)
2 (7)
4 (14)
6 (20)
Note: Numbers in parentheses are the percentages.
© 2014 International Union of Psychological Science
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TABLE 2
The number of children in each of the five choice patterns across the three motion problems
Category
Impetus-based
Inertia-based
Mixed
Straight-down
Other
Hovering
problem
Parallel movement
problem
Typical carried
object problem
Forward
Forward
T2
Forward
Straight down
T4 or T6
Forward
Forward
T3, T4, T5, or T6
Straight down
Straight down
T2
Choices in three problems could not be classified into any type
5-year-olds
8-year-olds
11-year-olds
13-year-olds
1 (3)
0 (0)
0 (0)
18 (58)
12 (39)
4 (13)
2 (7)
3 (10)
7 (23)
14 (47)
8 (28)
2 (7)
5 (17)
3 (10)
11 (38)
10 (33)
3 (10)
8 (27)
4 (13)
5 (17)
Note: Numbers in parentheses are the percentages.
children who chose the forward trajectories, 7 attributed
the bomb’s forward motion to impetus (e.g. “the bomb
will move forward because it has the force from the helicopter”). Only three children reported that the falling
bomb could move forward by itself. Other subjects simply
described the shape of the trajectories or reported that
their predictions were based on guesses.
Categories of participants’ choice patterns
across three problems
To understand a given child’s performance across the
three problems, each participant’s choices in the three
problems were summarised and classified into one of the
five categories (Table 2). The rationale of the categories
derives from existing formal physical models.
The mixed type
Children predicted the bomb to move forward in the
typical carried object problem (choosing T3, T4, T5 or
T6). However, they still chose the forward option in
parallel movement and the hovering problems. Both the
impetus and the inertia concepts could lead to this choice
pattern when air resistance was considered. Children’s
choice pattern alone could not distinguish the two possibilities, but older children’s verbal reports provided more
clues for distinction.
The straight down type
Children chose the downward options in all three
problems.
The other type
The impetus-based type
Children predicted the bomb to fall straight down
(choosing T2) in the typical carried object problem, and
chose to launch it forward to hit the moving tank in both
the parallel movement and the hovering problems. This
choice pattern is in accord with the prediction based on
the consistent engagement of the impetus concept: the
bomb itself cannot move forward after the release and
falls straight down regardless of the motion state of the
helicopter. Only being launched forward, can it obtain
forward impetus to catch up with the tank.
The inertia-based type
Children’s choices across the three problems were consistent with the inertia principle of Newtonian mechanics: (a) the bomb started to fall immediately after the
release and made inertial motion in the horizontal dimension (choosing T4 or T6);1 (b) due to inertia, the bomb in
the parallel movement problem just needed to be launched
downward to hit the tank and (c) the bomb should be
launched forward in the hovering problem.
The choice pattern across the three problems could not
be classified into any of the aforementioned types.
To focus on the developmental change of the
impetus-based type, it was compared with the other
four types combined. A 4 (age) × 2 (impetus-based vs.
non-impetus-based) chi-square test indicated that the proportion of the impetus-based type significantly increased
with age, χ2 (3, N = 120) = 10.96, p < .01, φ = .3. Five of
ten 13-year-olds in this type attributed the straight-down
trajectory chosen in the typical carried object problem to
the lack of impetus. The proportion of the mixed type also
significantly increased with age, χ2 (3, N = 120) = 10.06,
p < .01, φ = .29. Four of the eight 13-year-olds in this
type attributed the forward movement in the typical
carried object problem to the force obtained from the
helicopter. Very few children in the four age groups
belonged to the inertia-based type and no age difference
was found, χ2 (3, N = 120) = 3.21, p > .3, φ = .16.
There were 58% of 5-year-olds in the straight down
type, and the proportion significantly decreased with age,
χ2 (3, N = 120) = 22.48, p < .001, φ = .44. In addition,
quite a few children were categorised into the other
1 The difference between T4 and T6 is that the acceleration in the vertical dimension caused by gravity is considered in T6 but not T4. Given that
this difference is not the focus of the present study, the two trajectories were not strictly distinguished here.
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UNDERSTANDING OF PROJECTILE MOTION
type, χ2 (3, N = 120) = 6.52, p > .08, φ = .23. For example,
some of these children predicted the bomb to take forward trajectories in the typical carried object problem but
chose the downward option in the hovering problem.
DISCUSSION
Children of the four age groups showed quite different
response patterns across the three projectile problems.
Even though the three problems are from the same carried
object context, younger children’s trajectory predictions
were still influenced by contextual features of the motion
problems (i.e. the relative spatial locations of objects).
With the increase of age, children overcame the influence
of the spatial relationships among objects, and a proportion of 11- and 13-year-olds’ prediction patterns and
verbal reports suggest that these older children used the
impetus concept to predict projectile motion in the three
problems. Beyond simply emphasising children’s naïve
physics knowledge is coherent or piecemeal, this study
highlights the dynamic process in which children, with
age, gradually use the impetus concept across motion
problems.
This study does not provide direct evidence that
5-year-olds possess the impetus concept, although their
straight-down prediction in the typical carried object
problem suggests this concept may exist in this age.
The reason is that 5-year-olds understood that an object
without support would fall down and thus just focused
on the salient motion in the vertical dimension (Kaiser
et al., 1985). Without their verbal reports in the typical
carried object problem or other direct evidence, it is
difficult to conclude whether they really attributed the
object’s straight-down fall to the lack of forward impetus. Further studies are required to explore the early
emergence of the impetus concept in children. What this
study clearly demonstrates is that 5-year-olds’ trajectory
predictions were easily influenced by the relative spatial
relations between the helicopter and the tank (the parallel
movement and the hovering problems).
Although over half of 8-year-olds overcame the influence of spatial relationships of moving objects in the
hovering problem, they seemed to “regress” and use the
spatial relationships to make predictions in the more difficult parallel movement problem. In addition, 47% of
8-year-olds were in the other type and only 13% belonged
to the impetus-based type, suggesting that the majority of
8-year-olds, like 5-year-olds, did not engage the impetus
concept across the three problems.
In contrast, 11- and 13-year-olds’ performance differed from younger children’s in two aspects. First, more
11- and 13-year-old children made the forward movement prediction in the typical carried object problem,
consistent with previous findings with children of similar ages (Krist, 2000). This finding suggests that older
© 2014 International Union of Psychological Science
263
children might spontaneously revise their straight-down
belief during development to fit their observation of
the forward movement of falling objects in projectile
motion. However, this does not mean that these children
understood the Newtonian inertia principle, because
their verbal reports showed that about half of them
explicitly attributed the carried objects’ forward motion
to the impetus obtained from the carrier. This finding
also highlights that even though children do not show
the straight-down belief, they may still use the impetus
concept in predictions. Second, older children, more so
than younger ones, consistently engaged the impetus
concept to make trajectory predictions across the three
problems. Despite the fact that this study does not provide
direct evidence that children who were categorised in the
impetus-based type necessarily used the impetus concept,
the impetus concept engagement nevertheless provides
the most reasonable and parsimonious way to explain
this response pattern across the three problems.
Notably, a quarter of 11- and 13-year-old children
belonged to the mixed type, which could be led by
both the impetus-based and the inertia-based prediction
when air resistance was considered. However, half of
the 13-year-olds in this type reported that they attributed
the bomb’s forward movement to impetus. These children might assume that the bomb could continue moving
forward (choosing the forward movement in the typical
carried object problem), but the bomb gradually slowed
down due to air resistance or the dispassion of impetus during motion. Therefore, the bomb needed to be
launched forward to obtain extra impetus to catch up with
the tank. These children’s predictions in the mixed type
still seemed to be guided by the impetus concept.
Although a subset of older children engaged the impetus concept to make predictions across the present three
problems, it does not mean that they had built a coherent and well-developed impetus-based mental model
and used it across all projectile problems. This study
only included three motion problems within the carried
object context and did not manipulate various contextual
features or methodological factors that may influence
trajectory predictions (e.g. Anderson et al., 1992; Cooke
& Breedin, 1994; Howe et al., 2012). Particularly, only
children’s explicit conceptual knowledge, but not implicit
perceptual-motor knowledge, was examined in this study.
It would be interesting to investigate whether children
will show different responses if they are asked to execute
actions to solve the hovering and the parallel movement
problems. The findings will deepen our understanding of
the structure of children’s naïve physics knowledge.
When more contextual features, different types
of motion problems and different methodologies are
involved, it is not surprising to see that fewer children can
consistently engage the impetus concept in predictions
and more children will show piecemeal knowledge.
However, what is highlighted in this study is that, with
264
MOU, ZHU, CHEN
age, children organise their physical knowledge and
experience on the basis of some naive physical concepts
and gradually use the concepts to predict and interpret
more physical events (Baillargeon, 2008; Spelke, 1994;
Vosniadou, 1994; Vosniadou & Brewer, 1992).
In addition to improving our theoretical understanding of children’s naïve physics knowledge, the present
findings on the developmental changes of children’s
misconception are also beneficial to educational practice. Because younger children are highly susceptible to
contextual features in physical problems, school instruction can be designed to help them distinguish relevant
and irrelevant contextual features in problem solving. In
contrast, because 11- and 13-year-olds have already spontaneously and consistently engaged the impetus concept
to predict some motion problems before receiving formal
physics courses, instruction may be designed to correct their impetus concept to make effective conceptual
changes.
ACKNOWLEDGEMENTS
We thank Felicia Chu and Soek Jin’s comments on the
early version of the paper. We also thank Yunfei Bao, Hua
Chen, Guoping Liu, Wei Su, and Jing Yu for collecting
data for this study.
Manuscript received March 2014
Revised manuscript accepted July 2014
First published online September 2014
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