Design and Application of Interactive Simulations in Problem-Solving in University-Level Physics Education

Abstract

In recent years, interactive computer simulations have been progressively integrated in the teaching of the sciences and have contributed significant improvements in the teaching–learning process. Practicing problem-solving is a key factor in science and engineering education. The aim of this study was to design simulation-based problem-solving teaching materials and assess their effectiveness in improving students’ ability to solve problems in university-level physics. Firstly, we analyze the effect of using simulation-based materials in the development of students’ skills in employing procedures that are typically used in the scientific method of problem-solving. We found that a significant percentage of the experimental students used expert-type scientific procedures such as qualitative analysis of the problem, making hypotheses, and analysis of results. At the end of the course, only a minority of the students persisted with habits based solely on mathematical equations. Secondly, we compare the effectiveness in terms of problem-solving of the experimental group students with the students who are taught conventionally. We found that the implementation of the problem-solving strategy improved experimental students’ results regarding obtaining a correct solution from the academic point of view, in standard textbook problems. Thirdly, we explore students’ satisfaction with simulation-based problem-solving teaching materials and we found that the majority appear to be satisfied with the methodology proposed and took on a favorable attitude to learning problem-solving. The research was carried out among first-year Engineering Degree students.

Appendix

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A driver using his mobile phone suddenly notices that the traffic in front of him has stopped and he decides to brake, locking the wheels and skidding along the tarmac. Will a collision take place?

The teacher guided the students’ work with questions that aimed to direct scientific problem-solving. We will comment here on the most significant steps to solve the problem and we show the questions set by the teacher in italics:

How Can We Represent the Scenario We Have Depicted in the Statement in Graphic Form?

Comments: Some students do not understand the problem statements, and are unable to visualize the physical situation. The locking of the wheels that we have included in the scenario is a simplification which allows us to approach the problem with the knowledge available at the time when it is presented but does not actually reflect what usually happens with contemporary vehicles and so it may not be something the students will grasp intuitively. Watching the animation aids comprehension of the physical situation at the outset and helps to depict it visually in an appropriate way by means of a simplified model. In Fig. 1, we present an initial image of the interactive simulation that we designed to tackle this problem http://www.sc.ehu.es/sbweb/iem/vehiculo/vehicle.html. In this first view of the simulation we use the conditions at the outset which appear by default and in those all one sees is the movement of the vehicle involved in the possible collision.

Fig. 1

First view of the simulation to facilitate the description of the physical situation

What Magnitude Should We Calculate to Conclude Whether the Vehicle Collides or Not?

Comments: The first way of operationalizing the problem is to consider the distance, d, the vehicle covers from when the obstacle is detected to when it stops (stopping distance). If this distance is less than distance D to the traffic jam at the moment when the driver notices it, the vehicle will not collide.

Another way of approaching the problem is to think in terms of speed or in other words, if the speed, v, at which the car reaches the traffic jam is higher than zero, it will collide.

What Phases Should We Distinguish to Obtain the Stopping Distance?

Comments In accordance with the information provided in the statement, during the reaction time between using the accelerator and the brake, we could consider the vehicle’s speed to be constant, as long as it is assumed that the road is completely horizontal and that air resistance is negligible. Later, once the brake has been applied, the vehicle will reduce its speed with acceleration, a, that we consider to be constant, until it manages to stop. If the car skids on the tarmac with the wheels locked, there is no rolling movement. The stopping distance will be obtained by adding the distances covered during the reaction time (reaction distance, X _r) and the braking time (braking distance, d-X _r).

At this moment we can display the simulation of the most representative moments of the vehicle on the axis along which the rectilinear movement occurs and the diagram of the forces that are exerted on the vehicle. For the moment we would not display the controls. In this second playing of the simulation we can also analyze the graphics v(x) and a(x) corresponding to the movement in its various phases (see Fig. 2).

Fig. 2

Most representative moments of the movement, forces, and a(x) graph

Which Parameters Will Influence the Stopping Distance?

Comments: Making hypotheses play an essential role in problem-solving. Hypotheses focus and guide the problem-solving process, indicating the parameters to take into account. It is the hypotheses and the whole body of knowledge on which they are based that will allow us to analyze the result and the process as a whole.

It is also true that sometimes, even very often, students come up with misconceptions when they formulate hypotheses, but far from being negative, they may constitute the most efficient way of bringing to light alternative conceptions and lines of reasoning.

We attempt to answer the causal question “what magnitudes affect the stopping distance?” using reasoning. Taking into account that the stopping distance is the sum of the reaction distance and the braking distance, all the variables that influence these could affect the stopping distance. Therefore, it is possible that the initial speed, v ₀, the reaction time, t _r and the braking force affect the stopping distance. However, as the wheels are locked and the car is skidding, the braking force is a sliding friction force that depends on the kinetic friction coefficient, μ _k, on the mass, m, and the acceleration of gravity, g. It is therefore possible that d = d(t _r, v ₀, μ _k, g, m).

Following reflection by the students in groups on the various stages and which are the factors that could alter the distance necessary for the vehicle to come to a stop, a new simulation is run. We can show now the controls that will allow us to modify each of the relevant parameters (Fig. 3). The controls are not adjusted until there has been reflection on the next question.

Fig. 3

Visualization of the positions, forces exerted and parameters characteristic of the system

What Would Happen to the Stopping Distance If We Were to Modify the Magnitudes Given in the Previous Section, One by One, Keeping the Others Constant? You Should Give a Rationale for Your Hypothesis

Comments: With this question, posed to the students before adjusting the controls of the variables in the simulation, we aim to guide students to make theoretically founded hypothesis regarding the effect that modifying an independent variable would have on the dependent variable. We are attempting to justify how and why it is dependent in this way.

Due to space limitations, we will only present the hypotheses relating to the effects of two variables:

Hypothesis 1

If the stopping distance depends on the speed of the vehicle, v ₀, and the vehicle travels faster (keeping the remaining variables constant), the distance covered during the reaction time will increase and the braking distance will increase; then, the stopping distance should be greater.

Hypothesis 2

If the stopping distance depends on the vehicle mass, m, and the vehicle has more mass (keeping the remaining variables constant), the weight will increase, the normal force will increase, the braking force will increase and the braking distance will decrease but, in turn, the inertia will increase (it will take more effort to reduce the vehicle’s speed), and that will increase the braking distance; therefore, the stopping distance may not depend on the mass. (This double and opposite effect does not allow us to make a clear hypothesis of what will happen to the stopping distance if we vary the vehicle’s mass).

What Can We do to Check Whether Our Hypotheses Were Correct?

Comments: Once hypotheses have been made on variable dependency, it is necessary to test whether these hypothesis are confirmed. Solving the problem operatively allows us to derive an algebraic expression that relates the dependent variable to the independent variables. With this focus, problem-solving itself takes on a new meaning for students; as well as being used to answer the question posed in the statement, it constitutes an essential step to check the hypothesis made (and theoretically founded).

The kinematics of straight-line and uniform motion gives us the reaction distance. With Newton’s Second Law, we can find the braking acceleration that turns out to be constant given that the braking force is also constant. By means of straight-line motion with constant acceleration kinematics, we obtain the braking distance. The sum of these two distances will give us the stopping distance we are looking for:

$$d = v_{0} t_{r} + \left( {v_{0}^{2} /2\mu_{\text{k}} g} \right)$$

(1)

Are We Sure That Eq. (1) is Correct?

Comments: In order to check the validity of the result, we can try to solve the problem by using the kinetic energy theorem and check that we get the same result as we do if we use the kinematic–dynamic theory. In addition, we can carry out a dimensional analysis of the equation obtained and check that this is dimensionally homogeneous.

Are the Hypothesis Confirmed?

Comments: In the algebraic expression obtained for the stopping distance d, Eq. (1), it is observed that with regard to the velocity at which the vehicle is moving, v ₀, its increase makes the stopping distance longer, as we expected. It is noteworthy, however, that the braking distance (d − X _r) depends on the square of v₀, that is, going at twice the velocity means covering four times the distance from the moment when the brakes are activated to the moment when it comes to a halt. This is one of the variables that can be controlled by traffic police in order to reduce accidents.

With regard to the mass, m, for which we expected to contrary effects associated with the force of friction and the inertia, according to Eq. (1) these cancel each other out, and so in the event of hitting the brakes suddenly and the vehicle skids on the road surface (without rolling) neither the braking distance nor the stopping depends on the mass. This rather counterintuitive result is valid for the skidding circumstances used to model the problem but it is not valid if the force exerted by the brakes does not increase with the vehicle’s mass.

It is in this phase of testing our hypotheses where the interactive nature of the simulation demonstrates its explanatory power. We can change one by one all of the parameters that influence the braking distance while keeping the others constant, and observe how the animation shows us the consequent effect (see Fig. 4a, b). This allows us to assess the consistency between our hypotheses, the mathematical solution obtained for the problem by applying the laws and principles of physics and the physical phenomenon that really takes place.

Fig. 4

a Visualization of the simulation with values for the parameters that do not lead to a collision. b Visualization of the simulation with values for the parameters that do lead to a collision

Could We Apply the Results Obtained to Real Situations? At Times, We May Feel That the 50 km/h (31.1 mph) Speed Limit in Downtown and Urban Centers is Somewhat Excessive: Do You Think a 70 km/h (43.5 mph) Speed Limit Would be More Reasonable? At What Speed Would a Vehicle Crash into Another If the Driver Did Not Keep His/Her Distance?

Comments In this respect, we can encourage students to look for and/or estimate real data and use them to calculate and simulate the stopping distance. They can also calculate and simulate at what speed the collision would occur if the driver were not within a safe distance.

The reaction time for alert drivers ranges from 0.3 to 1.0 s. With a dry road surface and tires in good conditions, a good car can brake with a deceleration of between 5 and 8 m/s². Let us suppose, then, the following values:

$$t_{\text{r}} = 0.5\;{\text{s}}$$

$$a = \, \mu_{\text{k}} g = 6\;{\text{m}}/{\text{s}}^{2}$$

$$v_{0} = 50\;{\text{km}}/{\text{h}}\;{\text{and}}\;v_{{0^{\prime } }} = 70\;{\text{km}}/{\text{h}}$$

In the simulation, we observe that the stopping distance for 50 km/h is 23.05 m and for 70 km/h it comes out at 41.06 m. Therefore, at 70 km/h we would need more than 40 m to stop the vehicle if we had to due to some unforeseen circumstance, which, in downtown and urban centers means a high risk of an accident occurring—running someone over, for instance.

On the other hand, to illustrate the consequences of not keeping at a safe distance we can ask the students to consider a driving scenario in which the vehicle is moving at 100 km/h (62.1 mph), with a driver who has an average reaction time of 0.7 s and a vehicle with a braking capacity of 7.5 m/s². The vehicle behind makes an emergency stop, and from this point we can differentiate two scenarios:

• a. There is an appropriate safe distance between them: the driver notices the incident at a distance of 71 m or more and there is no crash.

• b. Distance between the vehicles is too short: the driver notices the incident at a distance of less than 71 m and crashes. If the driver becomes aware of the danger at a distance of 50 m, he will crash at a velocity of 64 km/h. If he realizes it at 28 m, the collision speed will be over 90 km/h.

After simulating these real situations, the students often expressed their surprise when they discovered that their ideas in terms of the appropriate speed limit and the safe distance between the cars were not consistent with the principles of safe driving.