Modelling

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Basic Modelling Skills

Justin learns in physics that a feather and brick dropped from the same height should reach the ground at the same time since the gravitational acceleration of all objects is equivalent, but he does not believe his teacher, Dr. Tyler. He decides to drop a feather and a brick from several heights and record the time it takes for them to reach the ground in a table:

Height (m)	0.2	0.4	0.6	0.8	1.0
Feather Time (s)	0.56	0.82	1.06	1.25	1.45
Brick Time (s)	0.20	0.29	0.35	0.40	0.45

Justin's experiment worked! The brick fell faster than the feather. After Justin brings his findings to Dr. Tyler, she tells Justin that he is right because the drag, or air resistance, force is high on the feather but very low on the brick. She admits that, in class, problems and examples tend to make the simplifying assumption that there is no friction or drag.

Discussion

In the absence of other forces - like drag - a dropped object's height, in meters, is

h(t)=h0−4.9t2,

where t is the time, in seconds, since it was dropped from an initial height of h0.

Based off of Justin's findings, explain whether you expect height of the brick or the feather to be more closely approximated by h.

The formula

h(t)=h0−4.9t2

is derived by assuming only gravity acts (no air resistance).

As Dr. Tyler explained, the feather took longer because the effect of drag on its motion is more noticeable. Therefore, the brick’s height will be much more closely approximated by h(t) than the feather’s since the effect of drag, which is excluded from h(t), is more significant on the feather than the brick.

Mathematical modelling and assumptions

A mathematical model is an equation or graph that represents a real-world situation and can be used to analyze and make predictions about that situation. Mathematical models may be exact or approximate.

Because real-world scenarios usually involve many variables, we often identify the most important ones and making reasonable assumptions about the rest. A good model simplifies the situation as much as possible without significantly reducing the accuracy of its predictions.

In a mathematical model, constants and coefficients are called parameters. The general shape of a model is given by its family (linear, quadratic, exponential, etc.), but the more specific values (like intercepts, asymptotes, or steepness) are controlled by the parameters.

The function h(t) might be considered a model for the time it takes the feather and the brick to reach the ground after they are dropped from a height h which makes the assumption that there is no air resistance. As we saw, h(t) is a poor model for the feather since the effect of air resistance was noticeable even from heights as low as 0.2m.

Discussion

Another student in Dr. Tyler's class, Hannah, models the temperature of ice as a constant heat is applied for her project. She plots the temperature C, in °C, versus time t, in minutes:

Describe a model - graphically or algebraically - that approximates Hannah's data.

The points plotted above appear to be scattered around a straight line.

Hence, a suitable model is a straight‐line fit through the endpoints (0,−10) and (5,−5). From these two points the slope is

m=5−0−5−(−10)=1

and the C-intercept is −10. Hence an algebraic model is

C(t)=t−10

Graphically, you would draw the line passing through (0,−10) and (5,−5) as the trend‐line for the scatter of data.

Linear models

A linear model is represented by a straight-line graph.

Since a linear model can be defined by one point and a gradient or two points, they are the simplest models to construct. The most common form of a linear model is y=ax+b, where a is the slope and b is the y-intercept.

Checkpoint

Using the model C(t)=t−10, predict the temperature of the ice after 12 minutes of heating.

Select the correct option

Discussion

Hannah continues monitoring the ice as it heats for 10 more minutes. Eventually, as the ice melts, the temperature differs from her predictions. The graph below presents Hannah's new data points in red and their distance from the expected values along the linear model C(t).

Until which time does it appear C(t) can produce accurate estimates of the temperature of the ice/water.

The measured temperatures coincide with the model C(t)=t−10 for 0≤t≤10. At t=11 and beyond the green dots lie below the line, so the model no longer matches the data. Therefore C(t) is accurate up to t=10 minutes.

Extrapolation

Extrapolation is when we predict values beyond the domain of the given points. Extrapolating may work for certain situations, but it does not work for many others. Pay attention to the context of a model when extrapolating and consider whether the observed behavior is likely to change in the long-run.

Your understanding of extrapolation can be tested by questions that ask you to interpret plausible inputs and outputs.

In the example with Hannah, we saw that extrapolating works but only up to a certain point.

Our initial model did not capture the transition from ice to water, so extrapolating to temperatures over the freezing point (0°C) did not work. However, up to those temperatures, we were able to extrapolate effectively.

In the context of this question, when considering how far you could extrapolate the model C(t), you should ask yourself questions like, "What will happen when the ice eventually melts?" If you do not know the answer to such a question, you should probably collect additional data rather than attempting to extrapolate.

Discussion

Hannah continues collecting data up to 25 minutes of heating. She plots all of her data and the original model C(t) but only up to t=10 (the values for which that model was accurate):

Explain graphically or algebraically how Hannah could add on to the existing model to create a model M(t) for the temperature of the ice/water of the first 25 minutes.
Describe how far we could reasonably extrapolate our new model. Consider if there is another temperature that you expect to behave differently than ice heating, ice melting, or water heating.

Part (a)

To extend the model for the temperature M(t) over the full 0≤t≤25 minutes, we need to account for three distinct phases, as seen in the data and the graph:

Cooling to 0∘C (ice only): For 0≤t≤10, the original model C(t) accurately describes the temperature as the ice is being heated up to its melting point.
M(t)=C(t)
Melting phase (temperature held at 0∘C): For 10<t≤15, the temperature remains constant at 0∘C while the ice melts. Here, 15 is the time when all the ice has just finished melting.
M(t)=0
Heating of water (after all ice has melted): For 15<t≤25, the temperature rises again as the water is heated. To model this, fit a new line through the point (15,0) and the measured temperature, approximately 5°C, at t=25. The equation of the line is:
M(t)=mt+b
where
m=25−155−0=21
and
b+15m=0⟹b=−15m=−215
This ensures the line passes through (15,0) and (25,5).

Graphically:

Plot C(t) from t=0 to t=10.
Draw a horizontal line at y=0 from t=10 to t=15.
From (15,0), draw the new heating curve (e.g., the straight line above) up to the point at t=25.

Combining these three segments gives a complete model M(t) for the temperature over the first 25 minutes.

Part (b)

Just as C(t) broke down once melting started, our final (water‐heating) line can only be trusted up to the point where the water reaches its boiling point (100°C) and evaporation begins.

We cannot assume that the temperature of the water will continue heating up at the same rate as it begins to evaporate.

Piecewise Linear Models

We use a piecewise linear model when different linear models apply over different parts of the domain of points. Basically, a piecewise linear model is a collection of smaller, domain-restricted models.

We write piecewise functions with the following notation:

f(x)={g(x),a<x<bh(x),b≤x<c

For example, we can model the temperature of ice/water in Hanna's experiment with

M(t)=⎩⎪⎨⎪⎧t−10,0≤t<100,10≤t<1521t−215,15≤t≤25

Exercise

Pick a reasonable piecewise model for the points plotted above.

Select the correct option

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Discussion

Jessica, a third student in Dr. Tyler's class, wants to model the relationship between the height of a tennis ball she throws into the air as a function of time. She plots her data on the graph below:

Describe a curve that could model the motion of the tennis ball. Note any graphical features of the curve that distinguish the model from a straight line.

A suitable model for the motion of the tennis ball is a smooth, symmetric curve shaped like an "upside-down U" (a parabola). This curve starts at the point (0,0), rises to a single maximum point near (1,5), and then descends back to (2,0), matching the pattern of the plotted data.

This curve is different from a straight line in several important ways:

The slope (rate of change) is positive for t<1, meaning the height increases as time increases.
At the top of the curve, around t≈1, the slope is zero. This is called a "turning point" or "vertex," where the ball reaches its maximum height before starting to fall.
For t>1, the slope is negative, so the height decreases as time increases.

A straight line would have a constant slope and could not model the ball's rise and fall. The presence of a single turning point and the change in slope are key features that distinguish this curve from a straight line.

Quadratic Models

A quadratic model has a turning point (vertex) at which its minimum or maximum value occurs. The general form of a quadratic equation is ax2+bx+c.

If a<0, the turning point of a quadratic is its maximum; if a>0, the turning point of a quadratic is its minimum.

Checkpoint

Fit the parabola to the plotted points.

Exercise

Find the minimum or maximum of −2x2+8x−5.

Select the correct option

Discussion

Jessica's model is reproduced above. Using the model, at which times was the ball on the ground?

The ball is on the ground whenever its height h=f(t) is zero, i.e. at the t–intercepts of the parabola. From the graph these occur at

t=0andt=2

so the ball is on the ground at t=0 s and t=2 s.

Quadratic x-intercepts

The roots of a quadratic correspond to the x-intercepts of its graph. When x=a or x=β, the entire expression equals zero, which is reflected on the graph.

The equation of the parabola below is −(x−α)(x−β):

Discussion

Jessica wants to study the key features of her model. She sketches a line along t=1, the x-value of the turning point of the parabola. Then, since she wants to check the two times when the ball reaches a certain height, Jessica draws a horizontal line from the point on the parabola at t=0.5 until it hits the other side of the parabola. She realizes that h(0.5)=h(1.5).

What can you deduce in general about the time(s) that the ball reaches a given height?

We first note two pairs of points at the same height:

h(0.5)=h(1.5)andh(0)=h(2)=0

in each case the two t-values are equidistant from t=1, the t-coordinate of the turning point.

Visually, the horizontal line at height h(0.5) and the base points (0,0) and (2,0) both look “mirror-image” about t=1 on the graph.

Algebraically, consider the simple parabola g(x)=x2. For any d>0,

g(d)=d2=g(−d)

so the points (d,d2) and (−d,d2) lie at the same height, equally spaced about the vertex at x=0. Our function h (or any parabola, for that matter) is just a combination of transformations applied to g, so equal heights occur at

t=1−dandt=1+d.

Hence in general, if the ball passes a height H first at time t, it passes the same height again at time

2−t,

or equivalently at

1−dand1+d

for some d.

Vertex and Axis of Symmetry

The graph of a quadratic function has the general shape of a parabola.

It is symmetrical about the axis of symmetry and has a maxima or minima at the vertex, which lies on the axis of symmetry.

Checkpoint

Find the axis of symmetry of the parabola graphed below.

Select the correct option

Exercise

It is given that a parabola has an x-intercept at (−3,0) and an axis of symmetry at x=−1. Find the other x-intercept of the parabola.

Select the correct option

Discussion

Hank forgot about his project and prepared no experiment to perform in class today! He remembers that he packed soup for lunch and decides to monitor its temperature every 2 minutes as it cools. He plots temperature C, in Celsius, against time t, in minutes:

Describe a curve that would model the temperature of Hank's soup.
Hank only monitored the temperature of the soup for an hour. Estimate the temperature of his soup when he eats it at t=240.

Part (a)

A suitable model is a smooth, steadily decreasing curve which drops quickly at first and then more slowly, flattening out towards a constant temperature. In particular:

– At t=0 the curve starts near 85∘C and falls steeply.

– As t increases the rate of cooling diminishes, so the graph “bends” and becomes less steep.

– By about t=40 min it is almost horizontal, approaching a horizontal asymptote at around 22∘C.

Note, this shape—rapid initial decrease, then gradual leveling off to a constant value—is characteristic of exponential decay.

Part (b)

Since in part (a) the curve “flattens” towards a horizontal asymptote at about 22∘C, and by t=60 min the temperature is already essentially 22∘C, extrapolating that flat portion to t=240 min gives

C(240)≈22∘C.

This makes sense physically: once the soup reaches room temperature it cools no further, so after four hours we still expect about 22∘C.

Exponential models

An exponential model represents quantities that multiply repetetively by a constant factor b. The basic form of an exponential is bx, but any exponential can be written in the form Abx+k.

The graph of an exponential model is a curve that approaches a horizontal asymptote at y=k on one side, and has a y-intercept at (0,A+k). Because of the asymptote on an exponential graph, exponential models are good at describing behaviors that level off over time.

Discussion

Given an exponential model f(x)=bx, explain for which values of b does f model
1. growth.
2. decay.

Part (a) (i)

For f(x)=bx to represent growth, we require that f increases as x increases. In particular, for any x,

f(x+1)>f(x).

But

f(x+1)=bx+1=b⋅bx>bx=f(x)

holds precisely when

b>1.

Hence f(x)=bx models exponential growth exactly for

b>1.

For example, 2x (doubling each step) and 1.5x (increasing by 50% each step) both grow because their bases exceed 1.

Part (a) (ii)

For decay we require that f decreases as x increases. In particular, for any x

f(x+1)<f(x).

But

f(x+1)=bx+1=b⋅bx<bx=f(x)

holds precisely when

b<1.

Moreover, to keep f(x)=bx from switching between positive and negatives values, we need b>0. Also b=1 gives the constant function f(x)=1, not decay. Hence the model decays exactly for

0<b<1.

For example, (21)x decreases as x grows, since its base 21 lies between 0 and 1.

Exponential growth

Exponential growth describes quantities that increase by the same factor over a certain amount of time. Algebraically, exponential growth are functions of the form

f(t)=Abt,

where b>1. b is called the growth factor.

Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is positive.

Stewart EJ, Madden R, Paul G, Taddei F (2005), CC BY-SA 4.0

Exponential decay

Exponential decay describes quantities that decrease by the same factor over a certain amount of time. Exponential decay are functions of the form

f(t)=Abt,

where 0<b<1. b is called the decay factor.

Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is negative.

Discussion

Amanda fixes magnets at positions x=−5cm and x=5cm. Then, she measures the force F applied to a third magnet at different positions. She plots her data on the graph below (negative positions and forces point toward Magnet 1, positive positions and forces point toward Magnet 2).

Describe a curve that could be used to model force as a function of position.

A good choice is a smooth, monotonically increasing S-shaped curve that goes through the points and has these visual features:

• At the left (around x≈–3 cm) it sits down near F≈–8 units and is quite steep.

• As you move toward the centre, the slope decreases, so by x≈–1 cm it’s already flattening off.

• Between x≈–1 and x≈1 cm the curve is almost flat, passing through (0,0) with a gentle incline.

• Beyond x≈1 cm the slope picks up again, becoming quite steep by x≈3 cm where F≈8 units.

• The result is one continuous “S” shape, negative and getting steeper to the left, inflecting near x≈–1, nearly straight through the middle, then again near x≈1 to become steeper up to the right.

Cubic Models

Cubic models have the form ax3+bx2+cx+d. Cubic graphs may have 0 or 2 turning points. When cubic graphs have 0 turning points, they have a short flat section where the function appears constant.

Discussion

Robert swings a pendulum and plots its height h, in centimeters, versus time t in seconds:

Describe how a model for the height of the pendulum as a function of time would look.

The graph of height versus time would be a single, smooth curve passing through the given points and then repeating its shape over and over. Visually, starting at the initial high point, the curve would sweep down to the low point, then rise back up to a high point, fall again, and so on in a regular, wave‐like pattern.

There are no sharp corners or breaks—just a continuous “up–down–up–down” motion of fixed size and time.

To capture that periodic, oscillating behavior one often uses functions that rise and fall smoothly—most commonly sine or cosine curves—to model the height of a pendulum over time.

Sinusoidal Models and their features

Sinusoidal models describe quantities that repeat in regular intervals, or periodically, and are typically of the form y=asin(bx)+c or y=acos(bx)+c.

A sinusoidal curve y=acos(bx)+c is graphed below with key features.

The principal axis, the line around which the sinusoid oscilates, is given by y=c.

The amplitude, or the maximum distance the sinusoid reaches above and below the principal axis, is a.

The period, or the horizontal distance between consecutive maxima, is given by b360° (or b2πrad for HL).

Exercise

Pick a suitable model for the points.

Select the correct option

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