3.3.2
Modelling with Parametric Equations
Modelling with Parametric Equations
Modelling with Parametric Equations
Parametric equations are used to model real-life situations.
![Illustrative background for Models](https://image-v2.cdn.app.senecalearning.com/2020-06/f484912a-0fe8-43d0-8dfb-550725eabfe8/rollercoaster,h_400,q_80,w_640.jpg)
![Illustrative background for Models ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/f484912a-0fe8-43d0-8dfb-550725eabfe8/rollercoaster,h_400,q_80,w_640.jpg)
Models
Models
- A mathematical model is a set of equations that describe a real-life situation or object.
- Answering a question about a model that uses parametric equations will require you to give your answer in units that are relevant to the question.
- You may also have to comment on how realistic a model is in the context of the question.
![Illustrative background for Time](https://image-v2.cdn.app.senecalearning.com/courseImages/physics/1.1.1 Measuring length, volume, etc.../clock-997589_640-min,h_400,q_80,w_640.jpg)
![Illustrative background for Time ?? "content](https://image-v2.cdn.app.senecalearning.com/courseImages/physics/1.1.1 Measuring length, volume, etc.../clock-997589_640-min,h_400,q_80,w_640.jpg)
Time
Time
- We can define a set of parametric equations with time as the parameter.
- Then and are defined as functions of time that describe the two-dimensional motion of an object.
![Illustrative background for Example](https://image-v2.cdn.app.senecalearning.com/2020-06/66cd0cf0-fcce-4760-89e4-a0dca9eecfea/Screenshot%202020-06-03%20at%2009.51.41-min,h_400,q_80,w_640.png)
![Illustrative background for Example ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/66cd0cf0-fcce-4760-89e4-a0dca9eecfea/Screenshot%202020-06-03%20at%2009.51.41-min,h_400,q_80,w_640.png)
Example
Example
- A remote-controlled boat is driven on the surface of a lake.
- The position (in metres) of the boat at time seconds is modelled by the parametric equations:
- Show that the motion of the remote-controlled boat is a straight line.
- Find the speed of the remote-controlled boat.
![Illustrative background for Rearrange $$x$$ equation](https://image-v2.cdn.app.senecalearning.com/2020-06/3df37e85-50e1-4065-b58c-7eed98e8f22e/Screenshot%202020-06-03%20at%2009.41.37-min,h_400,q_80,w_640.png)
![Illustrative background for Rearrange $$x$$ equation ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/3df37e85-50e1-4065-b58c-7eed98e8f22e/Screenshot%202020-06-03%20at%2009.41.37-min,h_400,q_80,w_640.png)
Rearrange equation
Rearrange equation
- To find the Cartesian equation, rearrange the equation for to get
on its own:
![Illustrative background for Find the Cartesian equation](https://image-v2.cdn.app.senecalearning.com/2020-06/f7ce2da8-a1e7-4c73-b7c2-4e15d28814bc/Screenshot%202020-06-03%20at%2009.41.39-min,h_400,q_80,w_640.png)
![Illustrative background for Find the Cartesian equation ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/f7ce2da8-a1e7-4c73-b7c2-4e15d28814bc/Screenshot%202020-06-03%20at%2009.41.39-min,h_400,q_80,w_640.png)
Find the Cartesian equation
Find the Cartesian equation
- Substitute this into the equation for gives:
![Illustrative background for Inspection](https://image-v2.cdn.app.senecalearning.com/2020-06/6144f3dc-32de-4ecb-9376-47c112122986/Screenshot%202020-06-03%20at%2009.41.41-min,h_400,q_80,w_640.png)
![Illustrative background for Inspection ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/6144f3dc-32de-4ecb-9376-47c112122986/Screenshot%202020-06-03%20at%2009.41.41-min,h_400,q_80,w_640.png)
Inspection
Inspection
- The equation is a straight line with gradient 3 and a -intercept of 0.
- and are the position of the boat, so the boat is moving in a straight line.
![Illustrative background for Find the speed of the boat](https://image-v2.cdn.app.senecalearning.com/2020-06/a06fd07d-26ff-46b1-a6cc-85d3e8d99a0e/Screenshot%202020-06-03%20at%2009.41.44-min,h_400,q_80,w_640.png)
![Illustrative background for Find the speed of the boat ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/a06fd07d-26ff-46b1-a6cc-85d3e8d99a0e/Screenshot%202020-06-03%20at%2009.41.44-min,h_400,q_80,w_640.png)
Find the speed of the boat
Find the speed of the boat
- Remember the equation for speed:
- Speed =
![Illustrative background for Find the speed in each direction](https://image-v2.cdn.app.senecalearning.com/2020-06/2af42d22-73a4-46e1-9b50-18d232c672a7/Screenshot%202020-06-03%20at%2009.41.46-min,h_400,q_80,w_640.png)
![Illustrative background for Find the speed in each direction ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/2af42d22-73a4-46e1-9b50-18d232c672a7/Screenshot%202020-06-03%20at%2009.41.46-min,h_400,q_80,w_640.png)
Find the speed in each direction
Find the speed in each direction
- In the -direction, the speed is:
- Speed = m/s
- In the -direction, the speed is:
- Speed = m/s
![Illustrative background for Find the magnitude of the speed](https://image-v2.cdn.app.senecalearning.com/2020-06/e9234ed7-e889-426b-860a-a8e1663752c8/Screenshot%202020-06-03%20at%2009.41.35-min,h_400,q_80,w_640.png)
![Illustrative background for Find the magnitude of the speed ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/e9234ed7-e889-426b-860a-a8e1663752c8/Screenshot%202020-06-03%20at%2009.41.35-min,h_400,q_80,w_640.png)
Find the magnitude of the speed
Find the magnitude of the speed
- The overall size of the speed of the boat is given by Pythagoras' theorem:
- Speed = m/s
- Remember to include units in your answers!
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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