Finding resultant vectors using a rubber band

Physics education researchers have found that most students have difficulty with vector addition and subtraction, both with and without a physical context [1–6]. In the case of graphical vector addition in both one and two dimensions, researchers have shown that most students still provide incorrect answers, even when they have studied the topic before [4–6]. One of the most crucial things they found was that some students made the attachment between two summed vectors correctly, using a 'tip-to-tail' strategy, but finally gave the wrong direction of the resultant vector. Most of those students drew the arrow of the resultant vector such that the triangle formed was 'flowing' in one direction [6]. We offer here an alternative way of finding a resultant vector with the aid of a rubber band, as described in the following.

A rubber band is cut into a straight line and one end is attached to a piece of paper cut in an arrowhead shape (see figure 1). This can be used to find the resultant vector from graphical vector addition and subtraction.

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To add any two or more vectors graphically, we start by drawing an initial vector while maintaining its magnitude and direction. Then, we connect the second vector's tail to the initial vector's tip—in so doing the magnitude and direction of the second vector should also be preserved. The same is done between the third vector and the second vector and so forth until all vectors are connected.

After the students have graphically made the attachment between two or more summed vectors, they can find the resultant vector by stretching the rubber band, following the connected vectors. This means holding one end of the rubber band fixed at the first vector tail and stretching the other end attached to the arrowhead paper, following each of the connected vector tips until it reaches the final vector tip. Figure 2 shows an example of how to use a rubber band to find the resultant vector of three vectors. This was done on an A4 sheet of paper.

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The stretched length of the rubber band represents the magnitude of the resultant vector and the arrowhead on the end of the rubber band points in the direction of the resultant vector.

To some observers, this seems to be not much different from drawing the resultant vector graphically by hand. However, with this method students do not become confused about the direction of the resultant vector. Moreover, the rubber-band method becomes useful when introducing the words distance and displacement to novice students in a physics course. A rubber band can be used to help them clarify these technical terms.

In a situation where an object moves from one position to another, the distance of the object is defined as the length of the actual path that the object has travelled while its displacement is defined as the change of the object's position. The displacement is a vector that points from the object's initial position to its final position and its magnitude is equal to the shortest distance between the two positions. A rubber band can obviously represent the displacement of an object. This is done by holding one end of the rubber band fixed at the object's first position, then stretching the arrowhead end following the object's travelled path and stopping at the object's final position. Figure 3(a) shows an example of an 'S'-shaped path travelled by an object. Figures 3(b)–(d) show how to use a rubber band to find the object displacement.

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Students can observe that the travelled path represents the object distance while the stretching of the rubber band immediately represents the object displacement. The stretched length reveals the magnitude and the arrowhead shows the direction of the displacement vector.

A rubber band is a very simple and cheap tool. It can be used to guide students to find a resultant vector easily. However, care must be taken when this is implemented in the classroom by students. Instructors should provide suitable rubber bands, with lengths such that they can be stretched safely. Careful supervision during implementation is also needed to prevent students using stretched rubber bands for fun.

The rubber band method can perhaps solve problems found in the literature [1–6]. By using a rubber band, students can clearly see the resultant vector. It helps them understand that the displacement is a vector that points from the object's initial position to its final position. They can also see changes of the displacement vector while the object is moving or changing its position. The tool helps novice students clarify the difference between the words 'distance' and 'displacement' as they are found in daily life and on their physics course.