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Date	May Specimen paper	Marks available	2	Reference code	SPM.2.AHL.TZ0.7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	7	Adapted from	N/A

Question

An object is placed into the top of a long vertical tube, filled with a thick viscous fluid, at time $t = 0$ seconds.

Initially it is thought that the resistance of the fluid would be proportional to the velocity of the object. The following model was proposed, where the object’s displacement, $x$ , from the top of the tube, measured in metres, is given by the differential equation

$\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)$ .

The maximum velocity approached by the object as it falls is known as the terminal velocity.

An experiment is performed in which the object is placed in the fluid on a number of occasions and its terminal velocity recorded. It is found that the terminal velocity was consistently smaller than that predicted by the model used. It was suggested that the resistance to motion is actually proportional to the velocity squared and so the following model was set up.

$\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9{\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)^2}$

At terminal velocity the acceleration of an object is equal to zero.

By substituting $v = \frac{{{\text{d}}x}}{{{\text{d}}t}}$ into the equation, find an expression for the velocity of the particle at time $t$ . Give your answer in the form $v = f(t)$ .

[7]

From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.

[2]

Write down the differential equation as a system of first order differential equations.

[2]

Use Euler’s method, with a step length of 0.2, to find the displacement and velocity of the object when $t = 0.6$ .

[4]

By repeated application of Euler’s method, find an approximation for the terminal velocity, to five significant figures.

[1]

Use the differential equation to find the terminal velocity for the object.

[2]

Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Euler approximation to this model.

[2]

Markscheme

$\frac{{{\text{d}}v}}{{{\text{d}}t}} = 9.81 - 0.9v$ M1

$\int {\frac{1}{{9.81 - 0.9v}}} {\text{d}}v = \int 1 \,{\text{d}}t$ M1

$- \frac{1}{{0.9}}{\text{ln}}\left( {9.81 - 0.9v} \right) = t + c$ A1

$9.81 - 0.9v = A{{\text{e}}^{ - 0.9t}}$ A1

$v = \frac{{9.81 - A{{\text{e}}^{ - 0.9t}}}}{{0.9}}$ A1

when $t = 0$ , $v = 0$ hence $A = 9.81$ A1

$v = \frac{{9.81\left( {1 - {{\text{e}}^{ - 0.9t}}} \right)}}{{0.9}}$

$v = 10.9\left( {1 - {{\text{e}}^{ - 0.9t}}} \right)$ A1

[7 marks]

either let $t$ tend to infinity, or $\frac{{{\text{d}}v}}{{{\text{d}}t}} = 0$ (M1)

$v = 10.9$ A1

[2 marks]

$\frac{{{\text{d}}x}}{{{\text{d}}t}} = y$ M1

$\frac{{{\text{dy}}}}{{{\text{d}}t}} = 9.81 - 0.9{y^2}$ A1

[2 marks]

${x_{n + 1}} = {x_n} + 0.2{y_n}$ , ${y_{n + 1}} = {y_n} + 0.2\left( {9.81 - 0.9{{\left( {{y_n}} \right)}^2}} \right)$ (M1)(A1)

$x = 1.04$ , $\frac{{{\text{d}}x}}{{{\text{d}}t}} = 3.31$ (M1)A1

[4 marks]

3.3015 A1

[1 mark]

$0 = 9.81 - 0.9{\left( v \right)^2}$ M1

$\Rightarrow v = \sqrt {\frac{{9.81}}{{0.9}}} = 3.301511 \ldots \,\,\left( { = 3.30} \right)$ A1

[2 marks]

the model found the terminal velocity very accurately, so good approximation R1

intermediate values had object exceeding terminal velocity so not good approximation R1

[2 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5—Calculus » AHL 5.14—Setting up a DE, solve by separating variables

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SPM.2.AHL.TZ0.7e:
By repeated application of Euler’s method, find an approximation for the terminal velocity, to five significant figures.
SPM.2.AHL.TZ0.7g:
Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Euler approximation to this model.
SPM.2.AHL.TZ0.7c:
Write down the differential equation as a system of first order differential equations.
SPM.2.AHL.TZ0.7b:
From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.
16N.2.AHL.TZ0.H_6:
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Give your answers in standard form.
SPM.2.AHL.TZ0.7a:
By substituting $v = \frac{{{\text{d}}x}}{{{\text{d}}t}}$ into the equation, find an expression for the velocity of the particle at time $t$ . Give your answer in the form $v = f(t)$ .
21M.3.AHL.TZ2.2d.ii:
Show that the expression for the mackerel population density after $t$ years is $M = M_{0} e^{α t}$
21M.1.AHL.TZ1.12b:
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21M.1.AHL.TZ1.12a:
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21M.3.AHL.TZ2.2d.i:
Write down the differential equation for $M$ that models this situation.
21M.3.AHL.TZ2.2d.iii:
Alessia estimates that the mackerel population density increases by a factor of three every two years. Show that $α = 0.549$ to three significant figures.
SPM.2.AHL.TZ0.7d:
Use Euler’s method, with a step length of 0.2, to find the displacement and velocity of the object when $t = 0.6$ .
21M.2.AHL.TZ1.7a:
Find the population of rabbits $1$ year after they were introduced.
21N.3.AHL.TZ0.2a.i:
Find the equation of the regression line of $h$ on $t$ .
21N.3.AHL.TZ0.2a.ii:
Interpret the meaning of parameter $a$ in the context of the model.
21N.3.AHL.TZ0.2a.iii:
Suggest why Eva’s use of the linear regression equation in this way could be unreliable.
21N.3.AHL.TZ0.2b.i:
Find the equation of the least squares quadratic regression curve.
21N.3.AHL.TZ0.2b.ii:
Find the value of $k$ .
21N.3.AHL.TZ0.2b.iii:
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21N.3.AHL.TZ0.2c:
Show that $\frac{d h}{d t} = - R^{2} \sqrt{70 560 h}$ .
21N.3.AHL.TZ0.2d:
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21N.3.AHL.TZ0.2e:
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21N.3.AHL.TZ0.2f:
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21N.3.AHL.TZ0.2g.ii:
Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$ .

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Topic 5—Calculus » AHL 5.18—Eulers method for 2nd order DEs

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