Modelling assumptions
In a nutshell
Mathematical models can be created to simulate real life scenarios involving motion and forces on objects. When creating these models, assumptions can be used to simplify them.
Constructing a model
When constructing mathematical models, it can be necessary to make certain assumptions so that it can be described using equations or graphs to solve them. Solutions to mathematical models must be interpreted in the context of the initial question, therefore your models may need to be refined using different assumptions.
procedure
| 1. | Identify the real life scenario.
|
| 2.
| Set up a mathematical model, with the assumptions and variables. |
| 3. | Solve the equation.
|
| 4. | Consider if your answer is reasonable contextually. If not, go to step 2 and change the variables. If yes, go to step 5. |
| 5. | Record the solution. |
Example 1
The motion of a football as a player kicks it can be modelled with h=10x−x2−4, where h is the height of the ball above the ground and x is the horizontal distance the ball travels.
a: Find the maximum height the ball reaches.
b: Predict the height of the ball when it is 10 m away from the player, and comment on the validity of this value.
a: The maximum height of the ball can be found by finding the turning point of the function:
dxdh=0
dxdh=10−2x=0
Solve for x:
10−2x=0
2x=10
x=5
Substitute this value for x into the intial equation:
h=10(5)−(52)−4=21
The maximum height of the ball is 21 m.
b: When x=10:
h=10(10)−(102)−4=−4
When the ball is 10 m away from the player, it has a height of −4 m.
The value for h is not sensible as the football cannot go below the ground. The value is invalid.
Modelling assumptions
Modelling assumptions can simplify problems and allow you to analyse real life scenarios using mathematical methods. Different assumptions can affect calculations differently, so it is important to know the assumptions and the way they affect models.
model | assumptions
|
|---|
| Particle | - Dimensions of the object are negligible.
- Rotational forces and air resistance can be ignored.
- Mass of the object is concentrated at one point.
|
| Rod
| - No thickness.
- Mass is distributed along a straight line.
- Does not bend or buckle.
|
| Lamina
| - Object has an area but negligible thickness.
- Mass is distributed across a flat surface
|
| Uniform Body
| - Mass is concentrated at the centre of the body - the centre of mass.
|
| Light Object
| - Objects mass is negligible.
- Tension is the same at both ends.
|
| Inextensible String
| - String does not stretch when carrying a load.
- Acceleration is the same in objects connected by an inextensible string.
|
| Smooth Surface
| - No friction between the surface and the object on it.
|
| Rough Surface
| - Objects moving on this surface experience frictional force.
|
| Wire
| - Treated as being one-dimensional.
|
| Smooth/light Pulley
| - No mass.
- Tension is the same on both sides of the pulley.
|
| Bead
| - Moves freely along a wire or string.
- Tension is the same on both sides.
|
| Peg
| - Fixed and dimension is negligible.
|
| Air Resistance
| - Acts in the opposite direction to movement.
- Usually negligible.
|
| Gravity
| - Gives objects a constant downwards acceleration of 9.8ms−2.
|
Example 2
Simon pulls a string on one side of a smooth pulley to pull up a weight with a mass of 60kg. What is the effect of the assumption that the weight is a particle and the pulley is smooth?
The assumption that the weight is a particle means that air resistance is ignored and the weight is concentrated at a single point. The assumption that the pulley is smooth means that tension is the same on either side of the pulley.
