Examples on Vector Addition

A car breaks down in the road and you and your friend, who happen to be walking past, help the driver push-start it. You and your friend stand together at the rear of the car. If you push with a force of \(\text{50}\) \(\text{N}\) and your friend pushes with a force of \(\text{45}\) \(\text{N}\), what is the resultant force on the car? Use the head-to-tail technique to calculate the answer graphically.

Step 1: Draw a rough sketch of the situation

Step 2: Choose a scale and a reference direction

Let's choose the direction to the right as the positive direction. The scale can be \(\text{1}\) \(\text{cm}\) = \(\text{10}\) \(\text{N}\).

Step 3: Choose one of the vectors and draw it as an arrow of the correct length in the correct direction

Start with your force vector and draw an arrow pointing to the right which is \(\text{5}\) \(\text{cm}\) long (i.e. \(\text{50}\) \(\text{N}\) = \(\text{5}\) \(\times\)\(\text{10}\) \(\text{N}\), therefore, you must multiply your \(\text{cm}\) scale by \(\text{5}\) as well).

Step 4: Take the next vector and draw it starting at the arrowhead of the previous vector.

Since your friend is pushing in the same direction as you, your force vectors must point in the same direction. Using the scale, this arrow should be \(\text{4.5}\) \(\text{cm}\) long.

Step 5: Draw the resultant, measure its length and find its direction

There are only two vectors in this problem, so the resultant vector must be drawn from the tail (i.e. starting point) of the first vector to the head of the second vector.

The resultant vector measures \(\text{9.5}\) \(\text{cm}\) and points to the right. Therefore the resultant force must be \(\text{95}\) \(\text{N}\) in the positive direction (or to the right).

Use the graphical head-to-tail method to determine the resultant force on a rugby player if two players on his team are pushing him forwards with forces of \(\vec{{F}_{1}} = \text{60}\text{ N}\) and \(\vec{{F}_{2}} = \text{90}\text{ N}\) respectively and two players from the opposing team are pushing him backwards with forces of \(\vec{{F}_{3}} = \text{100}\text{ N}\) and \(\vec{{F}_{4}} = \text{65}\text{ N}\) respectively.

Step 1: Choose a scale and a reference direction

Let's choose a scale of \(\text{0.5}\) \(\text{cm}\)= \(\text{10}\) \(\text{N}\) and for our diagram we will define the positive direction as to the right.

Step 2: Choose one of the vectors and draw it as an arrow of the correct length in the correct direction

We will start with drawing the vector \(\vec{{F}_{1}}= \text{60}\text{ N}\), pointing in the positive direction. Using our scale of \(\text{0.5}\) \(\text{cm}\) = \(\text{10}\) \(\text{N}\), the length of the arrow must be \(\text{3}\) \(\text{cm}\) pointing to the right.

Step 3: Take the next vector and draw it starting at the arrowhead of the previous vector

The next vector is \(\vec{{F}_{2}} = \text{90}\text{ N}\) in the same direction as \(\vec{{F}_{1}}\). Using the scale, the arrow should be \(\text{4.5}\) \(\text{cm}\) long and pointing to the right.

Step 4: Take the next vector and draw it starting at the arrowhead of the previous vector

The next vector is \(\vec{{F}_{3}} = \text{100}\text{ N}\) in the opposite direction. Using the scale, this arrow should be \(\text{5}\) \(\text{cm}\) long and point to the left.

Note: We are working in one dimension so this arrow would be drawn on top of the first vectors to the left. This will get messy so we'll draw it next to the actual line as well to show you what it looks like.

Step 5: Take the next vector and draw it starting at the arrowhead of the previous vector

The fourth vector is \(\vec{{F}_{4}} = \text{65}\text{ N}\) also in the opposite direction. Using the scale, this arrow must be \(\text{3.25}\) \(\text{cm}\) long and point to the left.

Step 6: Draw the resultant, measure its length and find its direction

We have now drawn all the force vectors that are being applied to the player. The resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector.

The resultant vector measures \(\text{0.75}\) \(\text{cm}\) which, using our scale is equivalent to \(\text{15}\) \(\text{N}\) and points to the left (or the negative direction or the direction the opposing team members are pushing in).