|Asia-Pacific Forum on Science
Learning and Teaching, Volume 4, Issue 2, Article 8 (Dec., 2003)
On using Geometer's Sketchpad to teach relative velocity
Example 2: To reach the other shore in the quickest possible time
Another problem that can be clearly illustrated by using GSP is to find the shortest time taken to cross the river. Suppose a current flows at a certain velocity w downstream. A boat is steered at a constant speed v. The direction the course taken is to be determined by the boat. The question is: what is the direction the boat should steer in order to reach the other side of the bank in the shortest possible time?
We construct the worksheet as in Figure 4 below. In Figures 4 and 5 below, the unit of measurement of time is in second.
The students are asked to drag the arrow, the direction of which represents the direction the boat is steered. Once decided on the direction the boat be steered, the button ANIMATE POINT on the screen can be clicked to demonstrate the path the actual boat travels (see Figure 5). The direction the boat travels represented by GE in Figure 5 will not be the same as the direction of steer of the boat, represented by GD. The point E represents the actual position of the boat while D is the position of the boat if there is no current downstream. Both points D and E have the same projection along the direction perpendicular to the two parallel river banks.
Students click and drag the course taken by the boat to allow for different choices of directions steered by the boat. By altering the course to be taken by the boat, the actual path traveled by the boat will be altered accordingly. This is immediately visible from the worksheet itself. The students can then be asked to explore on the course taken by the boat in order to reach the other shore in the quickest possible time by taking different courses in still water.
The students will discover that the course that the boat should take in order to reach the other shore in the quickest possible time is that perpendicular to the two parallel banks. The mathematical rigors of projection will be dispensed with while the objective is met.
As a bonus to the use of GSP in this case, the students will be able to make the following observations with the above file:
a. If the boatman steers upstream, the speed of the boat is slowed down compared with the speed he steers, and hence it takes a longer time to reach the other shore;
b. If the boatman steers downstream, the speed of the boat is increased from the speed he steers the boat (the further downstream, the faster the speed of the boat). However, the distance it has to take to cross the river is much longer, hence the time taken to reach the other shore is also increased.
Observations (a) and (b) above are difficult to demonstrate with the mechanical computation itself; with the use of such animation, the observations are quite visible to the students.
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