Vectors  (Page 5/10)

Its easy to understand the problem if we first draw a quick sketch. The rough sketch should include all of the information given in the problem. All of the magnitudes of the displacements are shown and a compass has been included as a reference direction. In a rough sketch one is interested in the approximate shape of the vector diagram.

The choice of scale depends on the actual question – you should choose a scale such that your vector diagram fits the page.

It is clear from the rough sketch that choosing a scale where 1 cm represents 2 km (scale: 1 cm = 2 km) would be a good choice in this problem. The diagram will then take up a good fraction of an A4 page. We now start the accurate construction.

Starting at the harbour H we draw the first vector 3 cm long in the direction north.

Take the next vector and draw it as an arrow starting from the head of the first vector in the correct direction and of thecorrect length.

Since the ship is now at port A we draw the second vector 6 cm long starting from point A in the direction east.

Take the next vector and draw it as an arrow starting from the head of the second vector in the correct direction and of thecorrect length.

Since the ship is now at port B we draw the third vector 2,25 cm long starting from this point in the direction south-west. A protractor is required to measure the angle of 45 ${}^{\circ }$ .

The resultant is then the vector drawn from the tail of the first vector to the head of the last. Its magnitude can bedetermined from the length of its arrow using the scale. Its direction too can be determined from the scale diagram.

As a final step we draw the resultant displacement from the starting point (the harbour H) to the end point (port C). We use aruler to measure the length of this arrow and a protractor to determine its direction.

We now use the scale to convert the length of the resultant in the scale diagram to the actual displacement in the problem. Since we have chosen a scale of 1 cm = 2 km in this problem the resultant has a magnitude of 9,2 km. The direction can be specified in terms of the angle measured either as $\mathrm{072,3}{}^{\circ }$ east of north or on a bearing of $\mathrm{072,3}{}^{\circ }$ .

The resultant displacement of the ship is 9,2 km on a bearing of $\mathrm{072,3}{}^{\circ }$ .

In the first part of his journey he traveled 40 m and in the second part he traveled 30 m. This gives us a total distance traveled of 40 m + 30 m = 70 m.

The man's resultant displacement is the vector from where he started to where he ended. It is the vector sum of his two separate displacements. We will use the head-to-tail method of accurate construction to find this vector.

A scale of 1 cm represents 10 m (1 cm = 10 m) is a good choice here. Now we can begin the process of construction.

We draw the first displacement as an arrow 4 cm long in an eastwards direction.

Starting from the head of the first vector we draw the second vector as an arrow 3 cm long in a northerly direction.

Now we connect the starting point to the end point and measure the length and direction of this arrow (the resultant).

To find the direction you measure the angle between the resultant and the 40 m vector. You should get about 37 ${}^{\circ }$ .

Finally we use the scale to convert the length of the resultant in the scale diagram to the actual magnitude of the resultantdisplacement. According to the chosen scale 1 cm = 10 m. Therefore 5 cm represents 50 m. The resultant displacement is then 50 m $37{}^{\circ }$ north of east.