Ash in the air

Students will also:

• use simple formulae
• work with areas of circles and ellipses
• interpret graphs
• learn to use everyday objects to simulate complex situations and understand how these models can be linked to the real situation.

Activity

The following practical demonstration that models the situation will help the students to appreciate what this problem is about. On a sheet of graph paper, mark a pair of axes as shown in the figure below. Note that the origin is the centre of the volcano.

Salt is an ideal substance for simulating ash. Use a good salt cellar with a single free-running hole, or perhaps a dry beaker from which the salt can be poured. Hold the salt container directly over the origin of the set of axes.

Pour the salt in an even stream for 1 minute and then carefully trace the outline of the salt pile onto the graph paper. Without disturbing the pile, pour the salt for a further 1 minute and trace around the pile again. Be sure that you pour the salt from the same height as before. Repeat the experiment for a further 3 or 4 minutes, tracing the outline of the pile each time.

Pour the salt back into the beaker and examine the outlines of the pile for each minute of salt pouring. Have the students calculate the area covered by the salt and record it in a table. The outline should be a circle, so the circle formula for area, A = πr², can be used.

This experiment models the distribution of the ash cloud in windless conditions. The example shown in the students’ book allows for a wind blowing from west to east. This can be modelled using the same equipment as before with the addition of a small fan. Begin with a fresh piece of graph paper and draw in the axes, with the volcano at the origin. Tape the paper down so that it doesn’t move. Now place a fan so that it blows across the paper as you pour the salt. Using the same method as before, draw around the “ash” for each minute of pouring. This time, the shapes will be elliptical and the area can be calculated using the formula A = πab, where a is the shortest possible radius measured from the centre and b is the longest possible radius.

Explain to the students that what has been done with salt would normally be modelled mathematically, using a computer.

When working on question 4, the students will find it helpful to sketch an approximation of the extent of the ash on the set of axes showing the towns. The required information comes from the two graphs. The westeast graph shows us that the spread starts at approximately 2.4 kilometres west of the volcano and stretches 14 kilometres to the east of the volcano. The centre of the ellipse is on the west-east axis at the point (5.8,0). The diagram below shows a bird’s-eye of the area covered by the ash.

In making an estimate of the depth of ash, allowance must be made for the fact that the towns do not lie on the west-east axis. If we take a north-south line through Chalet, for example, the ash will be thickest where this line crosses the west-east axis. The ash will be slightly shallower at Chalet and definitely shallower at Slalom and Titiromaunga.

Cross-curricular links

Social studies

The students could explore the advantages and disadvantages of new technology on the lives of people, for example, cellphones. They could debate a position on how these developments have impacted on their lives and the lives of others.

Achievement objectives

Demonstrate knowledge and understandings of:

• the impact of the spread of new technology and ideas on culture and heritage (Culture and Heritage, level 4)
• how and why people view and use resources differently and the consequences of this (Resources and Economic Activities, level 4)

Science

The students could investigate how instruments are used to predict the effect of the ash cloud. They could also discuss the effect of the ash cloud on the environment and the influence of weather patterns on the direction and impact of the ash cloud.

Achievement objectives

• investigate and offer explanations of how selected items of technology function and enhance everyday activities of people (Physical World, level 4)
• investigate the positive and negative effects of substances on people and on the environment (Material World, level 4)
• investigate major factors and patterns associated with weather, and use given data to predict weather (Planet Earth and Beyond, level 4)

See also: Building Science Concepts Book 12.

1.

A strong westerly wind is blowing the ash eastwards, creating an elliptical shape.

2.

The approximate areas are:

Looking at the differences, the growth appears to have stabilised at around 19 km² per hr, so 57 + 19 + 19 gives about 95 km² as a reasonable estimate of the area after 6 hrs.

3.

a. About 1.7 m deep

b. About 2 m deep

c. About 1 m deep

4.

a. Chalet, Slalom, and Titiromaunga need to be evacuated immediately.

b. Reasonable estimates are:

• Rockville and Skipton: no ash; Chalet: up to 3.5 m of ash; Slalom: about 1 m of ash; Titiromaunga: up to 0.8 m of ash.