12. Mathematical modelling in mechanics - Revision history

Ian at 18:41, 16 February 2010

2010-02-16T18:41:37Z

←Older revision		Revision as of 18:41, 16 February 2010
Line 87:		Line 87:
	So,		So,

-	<math>F=-\mu mg=-0.8\times 10m</math>	+	<math>F=-\mu mg=-0\textrm{.}8\times 10m</math>

	where		where
Line 108:		Line 108:
	& {{0}^{2}}={{u}^{2}}+2\times (-8)\times 20 \\		& {{0}^{2}}={{u}^{2}}+2\times (-8)\times 20 \\
	& {{u}^{2}}=320 \\		& {{u}^{2}}=320 \\
-	& u=17.9\text{ m}{{\text{s}}^{\text{-1}}} \\	+	& u=17\textrm{.}9\text{ m}{{\text{s}}^{\text{-1}}} \\
	\end{align}</math>		\end{align}</math>

Line 145:		Line 145:
	<math>\begin{align}		<math>\begin{align}
	& a=g(\mu \cos 5{}^\circ -\sin 5{}^\circ ) \\		& a=g(\mu \cos 5{}^\circ -\sin 5{}^\circ ) \\
-	& =9.8(0.8\cos 5{}^\circ -\sin 5{}^\circ ) \\	+	& =9\textrm{.}8(0\textrm{.}8\cos 5{}^\circ -\sin 5{}^\circ ) \\
-	& =-6.96\text{ m}{{\text{s}}^{\text{-2}}}	+	& =-6\textrm{.}96\text{ m}{{\text{s}}^{\text{-2}}}
	\end{align}</math>		\end{align}</math>

Line 153:		Line 153:

	<math>\begin{align}		<math>\begin{align}
-	& {{0}^{2}}={{u}^{2}}+2\times (-6.96)\times 20 \\	+	& {{0}^{2}}={{u}^{2}}+2\times (-6\textrm{.}96)\times 20 \\
-	& {{u}^{2}}=278.4 \\	+	& {{u}^{2}}=278\textrm{.}4 \\
-	& u=16.7\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 37 mph} \\	+	& u=16\textrm{.}7\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 37 mph} \\
	\end{align}</math>		\end{align}</math>

Line 167:		Line 167:

	<math>\begin{align}		<math>\begin{align}
-	& {{9}^{2}}={{u}^{2}}+2\times (-6.96)\times 20 \\	+	& {{9}^{2}}={{u}^{2}}+2\times (-6\textrm{.}96)\times 20 \\
-	& {{u}^{2}}=359.4 \\	+	& {{u}^{2}}=359\textrm{.}4 \\
-	& u=19.0\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 42 mph} \\	+	& u=19\textrm{.}0\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 42 mph} \\
	\end{align}</math>		\end{align}</math>

	It is interesting to note how close the two revised estimates are to the original prediction and that the conclusion that the car was breaking the speed limit was confirmed by both revisions.		It is interesting to note how close the two revised estimates are to the original prediction and that the conclusion that the car was breaking the speed limit was confirmed by both revisions.

Ian at 18:37, 16 February 2010

2010-02-16T18:37:01Z

←Older revision		Revision as of 18:37, 16 February 2010
Line 26:		Line 26:
	In the light of the review of the solution it will very often be desirable to refine the solution to the problem and the first stage in this will be to reformulate the problem to take account of new factors or to revise the assumptions.		In the light of the review of the solution it will very often be desirable to refine the solution to the problem and the first stage in this will be to reformulate the problem to take account of new factors or to revise the assumptions.
	The three stages of mathematical modelling are then often repeated until a satisfactory solution is obtained. The diagram shows how mathematical modelling has a cycle that can be repeated several times.		The three stages of mathematical modelling are then often repeated until a satisfactory solution is obtained. The diagram shows how mathematical modelling has a cycle that can be repeated several times.
		+

	[[Image:TF12.1.GIF]]		[[Image:TF12.1.GIF]]
Line 31:		Line 32:


-	~~'''~~An Example of Mathematical Modelling in Mechanics~~'''~~	+
		+	== An Example of Mathematical Modelling in Mechanics ==
		+
		+


Line 50:		Line 54:

	• Were all the wheels locked?		• Were all the wheels locked?
		+
		+	-------------------------------------

	Are there any further factors that you think should be included?		Are there any further factors that you think should be included?

	The assumptions below allow a simple model to be formulated so that a mathematical problem can be defined.		The assumptions below allow a simple model to be formulated so that a mathematical problem can be defined.
		+
		+	--------------------------------------

	• The road is horizontal.		• The road is horizontal.
Line 112:		Line 120:

	The Highway Code provides a useful source of data that can be used to compare the results with reality. The Highway Code quotes a distance of 80 feet or 24m as the braking distance for 40mph. This compares favourably with the prediction made above.		The Highway Code provides a useful source of data that can be used to compare the results with reality. The Highway Code quotes a distance of 80 feet or 24m as the braking distance for 40mph. This compares favourably with the prediction made above.
		+
		+
	'''Criticism of the Model'''		'''Criticism of the Model'''

Ian at 18:27, 16 February 2010

2010-02-16T18:27:02Z

←Older revision		Revision as of 18:27, 16 February 2010
Line 1:		Line 1:
-	12. Mathematical Modelling in Mechanics
-
	Mathematical modelling is used to describe the process of solving real world problems through the use of mathematics. The process begins when a problem statement passes through 3 main phases, formulation, solution and review, and culminates in a report. Mathematical models are often revised after the review stage and the formulation, solution, review process repeated. Each of these three phases is now considered in more detail.		Mathematical modelling is used to describe the process of solving real world problems through the use of mathematics. The process begins when a problem statement passes through 3 main phases, formulation, solution and review, and culminates in a report. Mathematical models are often revised after the review stage and the formulation, solution, review process repeated. Each of these three phases is now considered in more detail.

Line 11:		Line 9:

	The use of assumptions simplifies the problem considerably and allows a mathematical problem to be stated. The assumptions are important because they describe the conditions under which the mathematical solution will be obtained and they allow the problem to be simplified to an extent where it is possible to define a mathematical problem that can be solved.		The use of assumptions simplifies the problem considerably and allows a mathematical problem to be stated. The assumptions are important because they describe the conditions under which the mathematical solution will be obtained and they allow the problem to be simplified to an extent where it is possible to define a mathematical problem that can be solved.
		+

	'''Mathematical Solution'''		'''Mathematical Solution'''

	At this stage the mathematical problem defined in the formulation stage is solved to give a solution to the problem. This stage will involve many of the mathematical skills and techniques that you will have encountered in your mathematics and applied mathematics modules.		At this stage the mathematical problem defined in the formulation stage is solved to give a solution to the problem. This stage will involve many of the mathematical skills and techniques that you will have encountered in your mathematics and applied mathematics modules.
		+

	'''Review'''		'''Review'''
Line 28:		Line 28:

	[[Image:TF12.1.GIF]]		[[Image:TF12.1.GIF]]
		+
		+

	'''An Example of Mathematical Modelling in Mechanics'''		'''An Example of Mathematical Modelling in Mechanics'''
		+

	'''[[Problem Statement]]'''		'''[[Problem Statement]]'''
Line 41:		Line 44:

	• The road and tyre conditions.		• The road and tyre conditions.
		+
	• The gradient of the slope.		• The gradient of the slope.
		+
	• The speed of impact between the two cars. The conditions of the road.		• The speed of impact between the two cars. The conditions of the road.
		+
	• Were all the wheels locked?		• Were all the wheels locked?

Line 49:		Line 55:
	The assumptions below allow a simple model to be formulated so that a mathematical problem can be defined.		The assumptions below allow a simple model to be formulated so that a mathematical problem can be defined.

-	• The road is horizontal.	+	• The road is horizontal.
		+
	• The speed of impact is zero.		• The speed of impact is zero.
		+
	• There is constant friction force and no air resistance.		• There is constant friction force and no air resistance.
		+
	• The coefficient of friction between the tyres and the road is 0.8. The car is to be modelled as a particle.		• The coefficient of friction between the tyres and the road is 0.8. The car is to be modelled as a particle.
		+

	'''[[Mathematical Problem]]'''		'''[[Mathematical Problem]]'''

	A car skids 20m to rest on a horizontal road. If the coefficient of friction is 0.8 find the initial speed of the car.		A car skids 20m to rest on a horizontal road. If the coefficient of friction is 0.8 find the initial speed of the car.
-
-
-
-
-

	'''Mathematical Solution'''		'''Mathematical Solution'''
Line 74:		Line 79:
	So,		So,

-
	<math>F=-\mu mg=-0.8\times 10m</math>		<math>F=-\mu mg=-0.8\times 10m</math>
-

	where		where
Line 84:		Line 87:
	is taken as		is taken as

-	<math>10\ \text{m}{{\text{s}}^{-2}}</math>	+	<math>10\ \text{m}{{\text{s}}^{-2}}</math>.
-		+
-	.	+

	The acceleration of the car is given by		The acceleration of the car is given by
-

-	<math>a=\frac{F}{m}=-\frac{8m}{m}=-8\text{ m}{{\text{s}}^{\text{-2}}}</math>	+	<math>a=\frac{F}{m}=-\frac{8m}{m}=-8\text{ m}{{\text{s}}^{\text{-2}}}</math>.
-	.	+

	Now the speed can be found using		Now the speed can be found using

-
	<math>{{v}^{2}}={{u}^{2}}+2as</math>		<math>{{v}^{2}}={{u}^{2}}+2as</math>
-
-

	<math>\begin{align}		<math>\begin{align}
Line 111:		Line 107:

	The initial speed predicted can be converted to give 40mph. This figure clearly suggests that the car was breaking the speed limit.		The initial speed predicted can be converted to give 40mph. This figure clearly suggests that the car was breaking the speed limit.
		+

	'''Compare with Realitv'''		'''Compare with Realitv'''

	The Highway Code provides a useful source of data that can be used to compare the results with reality. The Highway Code quotes a distance of 80 feet or 24m as the braking distance for 40mph. This compares favourably with the prediction made above.		The Highway Code provides a useful source of data that can be used to compare the results with reality. The Highway Code quotes a distance of 80 feet or 24m as the braking distance for 40mph. This compares favourably with the prediction made above.
-
	'''Criticism of the Model'''		'''Criticism of the Model'''

	The two major criticisms that can be made of this model are that the road is not horizontal and that the speed of impact is not taken into account.		The two major criticisms that can be made of this model are that the road is not horizontal and that the speed of impact is not taken into account.

-	'''Reformulation'''

		+	'''Reformulation'''

	[[Image:TF12.3.GIF]]		[[Image:TF12.3.GIF]]
Line 129:		Line 125:
	to the horizontal. The normal reaction now has the magnitude,		to the horizontal. The normal reaction now has the magnitude,

-
	<math>R=mg\cos 5{}^\circ </math>		<math>R=mg\cos 5{}^\circ </math>
-

	and the resultant force up the slope on the car is,		and the resultant force up the slope on the car is,

-
	<math>\mu mg\cos 5{}^\circ -mg\sin 5{}^\circ =mg(\mu \cos 5{}^\circ -\sin 5{}^\circ )</math>		<math>\mu mg\cos 5{}^\circ -mg\sin 5{}^\circ =mg(\mu \cos 5{}^\circ -\sin 5{}^\circ )</math>
-

	So the acceleration is		So the acceleration is

-
	<math>\begin{align}		<math>\begin{align}
	& a=g(\mu \cos 5{}^\circ -\sin 5{}^\circ ) \\		& a=g(\mu \cos 5{}^\circ -\sin 5{}^\circ ) \\
Line 147:		Line 138:
	& =-6.96\text{ m}{{\text{s}}^{\text{-2}}}		& =-6.96\text{ m}{{\text{s}}^{\text{-2}}}
	\end{align}</math>		\end{align}</math>
-

	Using this value for		Using this value for
-	<math>a</math>	+	<math>a</math>with an impact speed of zero gives.
-	with an impact speed of zero gives.	+

-
	<math>\begin{align}		<math>\begin{align}
	& {{0}^{2}}={{u}^{2}}+2\times (-6.96)\times 20 \\		& {{0}^{2}}={{u}^{2}}+2\times (-6.96)\times 20 \\
Line 159:		Line 147:
	& u=16.7\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 37 mph} \\		& u=16.7\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 37 mph} \\
	\end{align}</math>		\end{align}</math>
-

	Notice that taking account of the hill reduces the initial speed		Notice that taking account of the hill reduces the initial speed
Line 168:		Line 155:
	or approximately		or approximately
	20 mph could be introduced. There was only minor damage to the vehicles which suggests that such a value is reasonable. Using this gives a revised initial speed		20 mph could be introduced. There was only minor damage to the vehicles which suggests that such a value is reasonable. Using this gives a revised initial speed
-

	<math>\begin{align}		<math>\begin{align}
Line 175:		Line 161:
	& u=19.0\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 42 mph} \\		& u=19.0\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 42 mph} \\
	\end{align}</math>		\end{align}</math>
-

	It is interesting to note how close the two revised estimates are to the original prediction and that the conclusion that the car was breaking the speed limit was confirmed by both revisions.		It is interesting to note how close the two revised estimates are to the original prediction and that the conclusion that the car was breaking the speed limit was confirmed by both revisions.

Ian at 16:13, 29 August 2009

2009-08-29T16:13:09Z

←Older revision		Revision as of 16:13, 29 August 2009
Line 4:		Line 4:


-	Formulation	+	'''Formulation'''

	Before embarking on the formulation stage it is important to make sure that you understand the problem that has been posed. The first part of the formulation stage is to make a list of all the features of the problem that could influence the solution you will eventually give.		Before embarking on the formulation stage it is important to make sure that you understand the problem that has been posed. The first part of the formulation stage is to make a list of all the features of the problem that could influence the solution you will eventually give.
Line 12:		Line 12:
	The use of assumptions simplifies the problem considerably and allows a mathematical problem to be stated. The assumptions are important because they describe the conditions under which the mathematical solution will be obtained and they allow the problem to be simplified to an extent where it is possible to define a mathematical problem that can be solved.		The use of assumptions simplifies the problem considerably and allows a mathematical problem to be stated. The assumptions are important because they describe the conditions under which the mathematical solution will be obtained and they allow the problem to be simplified to an extent where it is possible to define a mathematical problem that can be solved.

-	Mathematical Solution	+	'''Mathematical Solution'''

	At this stage the mathematical problem defined in the formulation stage is solved to give a solution to the problem. This stage will involve many of the mathematical skills and techniques that you will have encountered in your mathematics and applied mathematics modules.		At this stage the mathematical problem defined in the formulation stage is solved to give a solution to the problem. This stage will involve many of the mathematical skills and techniques that you will have encountered in your mathematics and applied mathematics modules.

-	Review	+	'''Review'''

	This stage contains a number of elements which include interpretation, comparison with reality, criticism of results and reformulation. In some cases it will be important for interpretation of the solution to be given.		This stage contains a number of elements which include interpretation, comparison with reality, criticism of results and reformulation. In some cases it will be important for interpretation of the solution to be given.
Line 27:		Line 27:
	The three stages of mathematical modelling are then often repeated until a satisfactory solution is obtained. The diagram shows how mathematical modelling has a cycle that can be repeated several times.		The three stages of mathematical modelling are then often repeated until a satisfactory solution is obtained. The diagram shows how mathematical modelling has a cycle that can be repeated several times.

		+	[[Image:TF12.1.GIF]]

		+	'''An Example of Mathematical Modelling in Mechanics'''

-		+	'''[[Problem Statement]]'''
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-	~~An Example of Mathematical Modelling in Mechanics~~	+
-		+
-	Problem Statement	+

	A car at an accident skids down a gentle slope leaving a skid mark of 20m before it collides with a stationary vehicle. How fast was the car travelling when it began to skid and was it breaking the 30mph speed limit?		A car at an accident skids down a gentle slope leaving a skid mark of 20m before it collides with a stationary vehicle. How fast was the car travelling when it began to skid and was it breaking the 30mph speed limit?

-	Formulation	+	'''Formulation'''

	The first step is to draw up a feature list to include all those factors that might affect the solution of the problem. The list below includes a number of important factors.		The first step is to draw up a feature list to include all those factors that might affect the solution of the problem. The list below includes a number of important factors.
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	• The coefficient of friction between the tyres and the road is 0.8. The car is to be modelled as a particle.		• The coefficient of friction between the tyres and the road is 0.8. The car is to be modelled as a particle.

-	Mathematical Problem	+	'''[[Mathematical Problem]]'''

	A car skids 20m to rest on a horizontal road. If the coefficient of friction is 0.8 find the initial speed of the car.		A car skids 20m to rest on a horizontal road. If the coefficient of friction is 0.8 find the initial speed of the car.
Line 85:		Line 63:


		+	'''Mathematical Solution'''

-		+	[[Image:TF12.2.GIF]]
-		+
-		+
-		+
-		+
-		+
-		+
-		+
-	~~Mathematical Solution~~	+

	As the car is skidding the friction force		As the car is skidding the friction force
Line 137:		Line 108:


-	Interpretation	+	'''Interpretation'''

	The initial speed predicted can be converted to give 40mph. This figure clearly suggests that the car was breaking the speed limit.		The initial speed predicted can be converted to give 40mph. This figure clearly suggests that the car was breaking the speed limit.

-	Compare with Realitv	+	'''Compare with Realitv'''

	The Highway Code provides a useful source of data that can be used to compare the results with reality. The Highway Code quotes a distance of 80 feet or 24m as the braking distance for 40mph. This compares favourably with the prediction made above.		The Highway Code provides a useful source of data that can be used to compare the results with reality. The Highway Code quotes a distance of 80 feet or 24m as the braking distance for 40mph. This compares favourably with the prediction made above.

-	Criticism of the Model	+	'''Criticism of the Model'''

	The two major criticisms that can be made of this model are that the road is not horizontal and that the speed of impact is not taken into account.		The two major criticisms that can be made of this model are that the road is not horizontal and that the speed of impact is not taken into account.

-	Reformulation	+	'''Reformulation'''
		+
		+
		+	[[Image:TF12.3.GIF]]

	The diagram shows the forces acting when the road is assumed to be at an angle of		The diagram shows the forces acting when the road is assumed to be at an angle of

Ian at 16:03, 29 August 2009

2009-08-29T16:03:28Z

←Older revision		Revision as of 16:03, 29 August 2009
Line 79:		Line 79:

	A car skids 20m to rest on a horizontal road. If the coefficient of friction is 0.8 find the initial speed of the car.		A car skids 20m to rest on a horizontal road. If the coefficient of friction is 0.8 find the initial speed of the car.
		+
		+
		+
		+
		+
		+
		+
		+
		+
		+
		+
		+
		+
		+

	Mathematical Solution		Mathematical Solution
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-	where m is the mass of the car and g is taken as 10 ms-2.	+	where
		+	<math>m</math>
		+	is the mass of the car and
		+	<math>g</math>
		+	is taken as
		+
		+	<math>10\ \text{m}{{\text{s}}^{-2}}</math>
		+
		+	.

	The acceleration of the car is given by		The acceleration of the car is given by
Line 129:		Line 151:
	Reformulation		Reformulation

-	The diagram shows the forces acting when the road is assumed to be at an angle of 5 to the horizontal. The normal reaction now has the magnitude,	+	The diagram shows the forces acting when the road is assumed to be at an angle of
		+	<math>5{}^\circ </math>
		+	to the horizontal. The normal reaction now has the magnitude,


Line 151:		Line 175:


-	Using this value for a with an impact speed of zero gives.	+	Using this value for
		+	<math>a</math>
		+	with an impact speed of zero gives.


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	Notice that taking account of the hill reduces the initial speed		Notice that taking account of the hill reduces the initial speed

-	As a final refinement to the model an impact speed 9 ms-l or approximately	+	As a final refinement to the model an impact speed
		+
		+	<math>9\ \text{m}{{\text{s}}^{-1}}</math>
		+	or approximately
	20 mph could be introduced. There was only minor damage to the vehicles which suggests that such a value is reasonable. Using this gives a revised initial speed		20 mph could be introduced. There was only minor damage to the vehicles which suggests that such a value is reasonable. Using this gives a revised initial speed

Ian: New page: 12. Mathematical Modelling in Mechanics Mathematical modelling is used to describe the process of solving real world problems through the use of mathematics. The process begins when a pro...

2009-08-29T15:30:21Z

New page: 12. Mathematical Modelling in Mechanics Mathematical modelling is used to describe the process of solving real world problems through the use of mathematics. The process begins when a pro...

New page

12. Mathematical Modelling in Mechanics

Mathematical modelling is used to describe the process of solving real world problems through the use of mathematics. The process begins when a problem statement passes through 3 main phases, formulation, solution and review, and culminates in a report. Mathematical models are often revised after the review stage and the formulation, solution, review process repeated. Each of these three phases is now considered in more detail.

Formulation

Before embarking on the formulation stage it is important to make sure that you understand the problem that has been posed. The first part of the formulation stage is to make a list of all the features of the problem that could influence the solution you will eventually give.

Some of the features identified may be very important while others are much more trivial. The second stage is to simplify the problem and to state a set of assumptions on which to base the problem.

The use of assumptions simplifies the problem considerably and allows a mathematical problem to be stated. The assumptions are important because they describe the conditions under which the mathematical solution will be obtained and they allow the problem to be simplified to an extent where it is possible to define a mathematical problem that can be solved.

Mathematical Solution

At this stage the mathematical problem defined in the formulation stage is solved to give a solution to the problem. This stage will involve many of the mathematical skills and techniques that you will have encountered in your mathematics and applied mathematics modules.

Review

This stage contains a number of elements which include interpretation, comparison with reality, criticism of results and reformulation. In some cases it will be important for interpretation of the solution to be given.

Comparison of results with real situations can be very useful for determining the validity of a solution.

Criticism of results is important to show any weaknesses of the solution, which may well be due to the initial assumptions.

In the light of the review of the solution it will very often be desirable to refine the solution to the problem and the first stage in this will be to reformulate the problem to take account of new factors or to revise the assumptions.
The three stages of mathematical modelling are then often repeated until a satisfactory solution is obtained. The diagram shows how mathematical modelling has a cycle that can be repeated several times.

An Example of Mathematical Modelling in Mechanics

Problem Statement

A car at an accident skids down a gentle slope leaving a skid mark of 20m before it collides with a stationary vehicle. How fast was the car travelling when it began to skid and was it breaking the 30mph speed limit?

Formulation

The first step is to draw up a feature list to include all those factors that might affect the solution of the problem. The list below includes a number of important factors.

• The road and tyre conditions.
• The gradient of the slope.
• The speed of impact between the two cars. The conditions of the road.
• Were all the wheels locked?

Are there any further factors that you think should be included?

The assumptions below allow a simple model to be formulated so that a mathematical problem can be defined.

• The road is horizontal.
• The speed of impact is zero.
• There is constant friction force and no air resistance.
• The coefficient of friction between the tyres and the road is 0.8. The car is to be modelled as a particle.

Mathematical Problem

A car skids 20m to rest on a horizontal road. If the coefficient of friction is 0.8 find the initial speed of the car.

Mathematical Solution

As the car is skidding the friction force
will take its limiting value of
<math>\mu R</math>
.

So,

<math>F=-\mu mg=-0.8\times 10m</math>

where m is the mass of the car and g is taken as 10 ms-2.

The acceleration of the car is given by

<math>a=\frac{F}{m}=-\frac{8m}{m}=-8\text{ m}{{\text{s}}^{\text{-2}}}</math>
.

Now the speed can be found using

<math>{{v}^{2}}={{u}^{2}}+2as</math>

<math>\begin{align}
& {{0}^{2}}={{u}^{2}}+2\times (-8)\times 20 \\
& {{u}^{2}}=320 \\
& u=17.9\text{ m}{{\text{s}}^{\text{-1}}} \\
\end{align}</math>

Interpretation

The initial speed predicted can be converted to give 40mph. This figure clearly suggests that the car was breaking the speed limit.

Compare with Realitv

The Highway Code provides a useful source of data that can be used to compare the results with reality. The Highway Code quotes a distance of 80 feet or 24m as the braking distance for 40mph. This compares favourably with the prediction made above.

Criticism of the Model

The two major criticisms that can be made of this model are that the road is not horizontal and that the speed of impact is not taken into account.

Reformulation

The diagram shows the forces acting when the road is assumed to be at an angle of 5 to the horizontal. The normal reaction now has the magnitude,

<math>R=mg\cos 5{}^\circ </math>

and the resultant force up the slope on the car is,

<math>\mu mg\cos 5{}^\circ -mg\sin 5{}^\circ =mg(\mu \cos 5{}^\circ -\sin 5{}^\circ )</math>

So the acceleration is

<math>\begin{align}
& a=g(\mu \cos 5{}^\circ -\sin 5{}^\circ ) \\
& =9.8(0.8\cos 5{}^\circ -\sin 5{}^\circ ) \\
& =-6.96\text{ m}{{\text{s}}^{\text{-2}}}
\end{align}</math>

Using this value for a with an impact speed of zero gives.

<math>\begin{align}
& {{0}^{2}}={{u}^{2}}+2\times (-6.96)\times 20 \\
& {{u}^{2}}=278.4 \\
& u=16.7\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 37 mph} \\
\end{align}</math>

Notice that taking account of the hill reduces the initial speed

As a final refinement to the model an impact speed 9 ms-l or approximately
20 mph could be introduced. There was only minor damage to the vehicles which suggests that such a value is reasonable. Using this gives a revised initial speed

<math>\begin{align}
& {{9}^{2}}={{u}^{2}}+2\times (-6.96)\times 20 \\
& {{u}^{2}}=359.4 \\
& u=19.0\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 42 mph} \\
\end{align}</math>

It is interesting to note how close the two revised estimates are to the original prediction and that the conclusion that the car was breaking the speed limit was confirmed by both revisions.