Determination of a car speed - analysis of projectile motion from a muddy puddle

Drops from a muddy puddle produce a distinct pattern on the side doors of a car driving through the puddle. This pattern was studied to determine the speed of the car. We can conceive at least two different models for pattern formation. We can decide in favor of one of the models by using different representations of the data. The activity is based on a rich context-based problem of projectile motion and offers epistemological insight into how physics research works by forming and testing hypotheses and using alternative representations.


Introduction
Projectile motion is a common phenomenon, easily observed in daily life.In introductory physics course, it is generally encountered after the study of one-dimensional kinematics, as an early example of two-dimensional motion [1].Projectile motion is taught in secondary schools but even university students show significant misconceptions concerning such motion [2].The traditional approach used in most textbooks is to define the magnitude and the angle from the horizontal of the projectile's initial velocity vector, then treat the motions in the horizontal and vertical directions independently.Uniform gravity is often the only force considered, so the equations of motion in the horizontal direction are for an object moving at a constant velocity, and the vertical direction equations are for an object undergoing a uniform acceleration.Analysis of the projectile motion regards the initial velocity, gravitational acceleration, launch angle, time of flight, the highest point of the trajectory, and the maximal range.Problems considered are often only an exercise in manipulating equations.As an alternative, all the expressions can be derived from the analysis of the graphs of motion [3].Some simple geometrical arguments and the power of symmetry can be used to analyse projectile motion in great generality, deriving the direction of velocity at any point, range, time of flight, maximum height, the safety parabola, and maximum range [4,5].Projectile motion is also a popular experimental example and is suited for video analysis [6], which has already been considered to tackle problems of misconceptions about projectile motion [7].In [8] a simple laboratory activity for introductory-level physics students is described.The activity allows students to confront their misconceptions about the conservation of energy and projectile motion and to clearly see the power of model predictions in understanding the behaviour of physical systems.An experiment using the volumetric flux of a water stream can also be used to find the initial velocity of water, comparing the maximum height and range values with the theoretical calculations [9].An interesting problem in locating the source of the projectile fluid droplets is analysed in [10].The selection and application of coordinate systems is also an important issue in physics.However, considering different frames of reference in a given problem sometimes seems unintuitive and difficult for students, and examples of projectile motion are welcome [11].The deterministic nature of simple physical systems is favourable in simple problems, however, in realistic systems, uncertainty is largely present via external forces such as friction or lack of precise knowledge about the initial conditions of the system.A classical system subject to uncertain initial conditions was described using inference methodologies in solving its statistical properties in [12].
Science teaching as content-driven lectures where abstract concepts are presented first and later illustrated with idealized examples, removed from personal experience and interest, is under growing criticism [13,14].In such an approach, memorization of facts and algorithmic problem-solving prevail over conceptual understanding.It is assumed new information fits in a pre-existing framework with all proper connections, supplanting any contradictory ideas the students may already hold.The structure of traditional science courses erects numerous roadblocks to students becoming actively involved in their learning.Encouraging students to remain in this passive role in the classroom has further unfortunate effects of promoting rote learning.Alternative instructional methods are thus encouraged and one of them is context-based physics instruction [15].The problems that are used to motivate and focus students' learning also connect previous knowledge in science to new concepts to be mastered.A crucial component of the problem in a problem-based learning environment is the introduction of concepts and principles.Good problems tell an engaging story to which the students can relate and thus solidify the connection between theory and application.If they are open-ended, they can challenge students to make and justify assumptions and estimates.Problems with controversial results require decisions and students must employ thinking skills beyond simple knowledge and comprehension to solve them.An example of a context problem for projectile motion is given in [16].An important aspect of teaching physics is also teaching students how to think like physicists [17].Instrumental in this task is the Investigative Science Learning Environment (ISLE) approach [18], which makes use of observational experiments, forming explanations based on observed patterns and subsequent testing.This paper describes an activity based on the ISLE approach regarding projectile motion where explanations of the observed pattern are put to the test.

Method
The ISLE approach is an intentional holistic learning environment that helps students learn physics by engaging them in a process that mirrors scientific practice.This approach involves students working in groups, developing their own ideas by observing phenomena and looking for patterns, developing explanations for these patterns, testing predictions of these explanations with experiments, deciding if the outcomes of the testing experiments are consistent with the predictions, and revising the explanations if necessary.Students also learn to represent results and descriptions in multiple ways.
Here we present a problem, which serves as an observational experiment and is posed as a problem in an everyday context.Students observe a pattern of muddy drops on the doors of a car shown in Figure 1 and must form at least two explanations, models for the observed pattern.They test the validity of the models by comparing their predictions to the measured values.The accepted model is used to determine the speed of the car at the time when the pattern was formed.The trajectory formula of a drop is given by where the  axis is horizontal and  is vertical, the origin is set at the point of the drop release,  0 is the drop initial velocity, and  is the angle of the initial velocity with the horizontal.Many drops end up on the car doors, each one following the ballistic trajectory.As a drop hits the surface, it produces a spatter tangential to the trajectory of the drop.A multitude of such streaks produces a pattern where a family of parabolas can easily be discerned, just like magnetic field lines when demonstrated with iron filings.There are several causes for many different parabolas.The car could be traveling at different speeds during the formation of the pattern.We assume, that the speed of the car was not changing during the time it took for it to travel through the puddle because the number of muddy drops is not exceedingly large meaning the wheel was in contact with the puddle for a relatively short time.Two other models of pattern formation can be considered.In one the drops leave the rim of the wheel at a random point, thus having a variable  and a fixed  0 .In this case,  0 equals the rim orbital speed in the car's reference frame which is the same as the speed of the car in the laboratory reference frame.
Variable  then accounts for the drops covering the whole side of the car and not flowing in a single stream.In another model for the muddy pattern formation, we consider the drops that are sent vertically with variable initial vertical speed , as the wheel splashes in the puddle.The horizontal component of their velocity relative to the car is fixed and given by the speed of the car.In the second model, the trajectory is given by where  is the variable vertical component of the drop velocity and the frame of reference is traveling with the car.The variable in the first model is  and in the second it is .These two variables are hard to discern in the muddy pattern on the car doors.An alternative way of describing the two-parameter free-fall parabolic pattern is using the height ℎ and range  of the parabolas corresponding to the pattern of the series of drops as shown in Figure 2. and  2 =  0 . For a given value of variable  (or ) the two parameters, height, and range, are not independent but related as and Thus, we have eliminated the unknown variables  and  from the problem.

Results
The data were gathered for several (nine) parabolas associated with muddy pattern on the doors.The range and height of parabolas are presented in the diagram in Figure 3

Discussion
With the activity proposed in this paper, students can follow the physicists thinking while evaluating measured data and can eventually arrive at the desired result, that is the speed of the car at the time of the pattern formation.The activity was tested with three groups: gifted end-of-primary school pupils during summer school, secondary school physics teachers, and graduate physics students.The first idea for the scattered muddy pattern from all the groups was a variable car speed.The small number of droplets leads to the realization that the contact time of the car with the puddle is short.The assumption of constant speed was readily adopted.All the groups were able to propose two different descriptions of pattern formation on their own.The students of the first group were not familiar with parabolic trajectories and theoretical guidance from the instructor was needed.Members of the other two groups had no difficulties arriving at the expressions for the two models.The students were individually able to calculate the car speed according to the two models but were unable to distinguish which model better suits the observed pattern.Here guidance from the instructor was again necessary for secondary and primary school students.The group of teachers was familiar with the graphlinearization concept.Finally, all groups were able to decide on the better model based on the suggested representations.All participants were highly motivated by an everyday context problem.Some teachers, familiar with ISLE approach, also expressed praise for the problem where 'crazy ideas' (as alternative descriptions of the observation [18]) were not too obviously false.The activity can be extended for the university student by employing more rigorous hypothesis testing methods such as  2 test.

Figure 1 .
Figure 1.A pattern of muddy drops on the side of a car that drove through a puddle.

Figure 2 .
Figure 2. The parabolic pattern of muddy drops on the side of the car can be characterized by its height ℎ and range .

Figure 3 .
Figure 3. (a) Measured data for several parabolic profiles in the muddy pattern together with the bestfit plots (equations (3) and (4)).(b) Linearised plot of the model of equation (3) and suitably rescaled data points.