Inquiry Based Teaching
3. Work with computer simulation “My Solar System”
The simulation MSS gives large possibility to work with students. There is a variety of tasks, from the simplest ones to complex tasks requiring independent investigative work of students. First of all, the students should be familiar with the program and learned to work with it. Students were
encouraged to examine this own "pocket universe". In the next section we will describe briefly how it could look like teacher work with students of guided scientific inquiry.
The instructions were as follows: Imagine that you have just created the universe and you have chosen appropriate units of time, mass and length, so you can examine its properties. We will start with 2 bodies of similar mass (20 M and 10 M, or 20M and 30 M, 10 M and 50 M and so on). It is advantageous if each group of students will select a different combination of masses. Place the bodies on the x-axis, for example into the points (-400, 0) L and (100, 0) L. Initial velocity of both bodies should be zero. Start the program and observe the movement. Reset and start again. Start and stop the motion repeatedly and measure their velocity and position in various stages of movement. Write down intermediate values of time, velocity and position of both bodies. It is preferable to write it into Excel form. Find the position of a point where the bodies collide. How the ratio of the speed of bodies depends on their masses? Calculate (preferably in Excel) the distance traveled by each of the bodies and find how their ratio depends on the masses. The example of student’s results is shown in Tab.1.
Expected conclusion of students, formulated in words or formula is that the ratio of velocities is equal to the reciprocal ratio of their masses, i.e.
1 2 2 1 m m v v (1) As a next step we will discuss students’ observations and conclusions. Edit the formula (1) so, that on the left side of the equation will be only variables related to the
first body and on the right side related to the second body. 2 2 1 1v m v m (2) t 0 6,9 10,9 13,6 15,3 16 m1 50 x1 -400,0 -390,1 -373,5 -355,3 -334,4 -317,5 v1 0 3 5,6 8,8 15,7 81,1 m2 10 x2 100,0 50,6 -32,4 -123,4 -227,8 -312,5 v2 0 -15 -27,9 -43,9 -78,4 -405,3 s1 0 9,9 26,5 44,7 65,6 82,5 s2 0 -49,4 -132,4 -223,4 -327,8 -412,5 s1/s2 -0,20 -0,20 -0,20 -0,20 -0,20 v1/v2 -0,20 -0,20 -0,20 -0,20 -0,20
Table 1. Measurement of velocities Now we can define some physical quantity describing the movement of the body as a product of its velocity and mass – we name it momentum H = m.v. What is the physical meaning of equation (2)? At any point of time is the momentum of first body the same size but opposite direction as the momentum of second body. The sum of both momentums is zero all the time. Therefore, after collision the resulting body will have zero momentum, and, as a result zero velocity.
How will change the results of the experiment, when the initial velocity of the first body will not be zero, and will have, for example, value of v0 2 2 0 1 1 1
v
m
v
m
v
m
= 10 L/T in x-axis direction? Trace the change in momentum over time. The results is(3)
So, step by step we will guide our students to the concept conservation of momentum, momentum as vector quantity and even to the concept of force and Newton's law of force. Why body changes its momentum? Because other body acts on it. We can define new physical quantity named force as a measure of acting of one body to other. Examine how quickly changes the momentum of the body. Repeat our first experiment with masses twice smaller. We can see that change of momentum is slower. Thus, it is reasonable to define force as a measure of change of momentum in time
t H F ' ' (4) The next situation, in which we can use MSS simulation, is measurement of acceleration and investigation how it depend on masses and mutual distance of bodies. Let choose the mass of first body m1 = 0.001 M and second body m2 = 100 M with initial velocities equal zero. As we know from first experiment, the velocity of second body will be 100 thousand times smaller than velocity of first one, and so we can neglect it. Second body will stay in its initial position. Place the first body into point (0, 0) L and second body into point (1000, 0) L. Run the program and stop it after few T. How we can measure initial acceleration of the first body? We expect that students know the formulas for uniformly accelerated motion and therefore they propose measuring of acceleration either from formula s = ½ a. t2 or from formula a = v/t (initial velocity is zero). Of course, it is necessary to discuss the precision of both methods. The students were encouraged to measure the initial acceleration of first body for various initial distances between both bodies (for example
600 L, 800 L, 1000 L and so on). The example of student’s results is shown in Tab.2. x2 t x v a=v/t a=2x/t2 R a.R2 600 5,5 43 16,1 2,927 2,843 578 965542 800 7 38,8 11,3 1,614 1,584 781 974316 1000 8,9 40,5 9,2 1,034 1,023 980 986934 1200 10,5 38,3 7,4 0,705 0,695 1181 975769 1400 12,2 38,1 6,3 0,516 0,512 1381 980546 1600 17,4 59,5 7 0,402 0,393 1570 980541 1800 19,4 58,6 6,1 0,314 0,311 1771 981118 2000 21,9 60,8 5,6 0,256 0,254 1970 987767 2200 24,1 60,4 5,1 0,212 0,208 2170 987754 2400 25,6 57,3 4,5 0,176 0,175 2371 985897 2600 27,9 57,8 4,2 0,151 0,149 2571 988428 2800 29,6 56,4 3,8 0,128 0,129 2772 987718 3000 33,2 61,7 3,7 0,111 0,112 2969 984729
Table 2. Measurement of acceleration Expected conclusion of students is that both method of the acceleration measurement give very similar values. Acceleration is proportional to mass of second body and inversely proportional to the square of its distance. Distance R between the bodies was calculated as an average value of initial and final distances, i.e. R = x2
2 2 1 R m G a – ½ x. This dependence we can write by formula as
(5) where constant of proportionality G calculated from the last column of the Tab.2 has value G = 9821 L3M-1T-2
The simulation MSS gives us a lot of other possibilities for guided scientific inquiry. For example, from the values in Tab.1 we can calculate the kinetic energy of bodies and show, that sum of the kinetic energy depend on mutual distance of bodies as follows:
. From this result we can easily deduce the gravitational law. R K E Ekin 0 (6) where E0 = -9906 ML2T-2 and K = 4,978 x 106 ML3T-2. From this result we can easily obtain formula for potential energy in gravitational field and also gravitational constant.
4. Computer simulation as a testing