Constant & Variable Velocity | DP IB Maths: AI HL Revision Notes 2021

Vectors & Constant Velocity

How are vectors used to model linear motion?

If an object is moving with constant velocity it will travel in a straight line
For an object moving in a straight line in two or three dimensions its velocity, displacement and time can be related using the vector equation of a line

r = a + λb
Letting

r be the position of the object at the time, t
a be the position vector, r₀ at the start (t = 0)
$λ$ represent the time, t
b be the velocity vector, v

Then the position of the object at the time, t can be given by

r = r₀+ tv

The velocity vector is the direction vector in the equation of the line
The speed of the object will be the magnitude of the velocity |v|

Worked example

A car, moving at constant speed, takes 2 minutes to drive in a straight line from point A (-4, 3) to point B (6, -5).

At time t, in minutes, the position vector (p) of the car relative to the origin can be given in the form $p = a + t b$ .

Find the vectors a and b.

3-10-2-ib-aa-hl-kinematics-vectors-we-solution

Vectors & Variable Velocity

How are vectors used to model motion with variable velocity?

The velocity of a particle is the rate of change of its displacement over time
In one dimension velocity, v, is found be taking the derivative of the displacement, s, with respect to time, t
- $v = \frac{d s}{d t}$
In more than one dimension vectors are used to represent motion
For displacement given as a function of time in the form
- $r (t) = (\binom{f_{1} (t)}{f_{2} (t)})$
The velocity vector can be found by differentiating each component of the vector individually
- $v = (\binom{v_{1} (t)}{v_{2} (t)})$
- $v = \frac{d r}{d t} = (\binom{f_{1}' (t)}{f_{2}' (t)})$
- The velocity should be left as a vector
- The speed is the magnitude of the velocity
If the velocity vector is known, displacement can be found by integrating each component of the vector individually
- The constant of integration for each component will need to be found
The acceleration of a particle is the rate of change of its velocity over time
In one dimension acceleration, a, is found be taking the derivative of the velocity, v, with respect to time, t
- $a = \frac{d v}{d t} = \frac{d^{2} r}{d t^{2}}$
In two dimensions acceleration can be found by differentiating each component of the velocity vector individually
- $a = (\binom{a_{1} (t)}{a_{2} (t)})$
- $a = \frac{d v}{d t} = (\binom{v_{1}' (t)}{v_{2}' (t)})$
- $a = \frac{d^{2} r}{d t^{2}} = (\binom{f_{1}'' (t)}{f_{2}'' (t)})$
If the acceleration vector is known, the velocity vector can be found by integrating each component of the acceleration vector individually
- The constant of integration for each component will need to be found

Examiner Tip

Look out for clues in the question as to whether you should treat the question as a constant or variable velocity problem
- 'moving at a constant speed' will imply using a linear model
- an object falling or rolling would imply variable velocity

Worked example

A ball is rolling down a hill with velocity $v = (5 3) + t (0 - 0.8)$ . At the time t = 0 the position vector of the ball is 3i-2j.

Find the acceleration vector of the ball's motion.

3-9-2-ib-ai-hl-variable-velocity-we-solution-a

Find the position vector of the ball at the time, t.

3-9-2-ib-ai-hl-variable-velocity-we-solution-b

Constant & Variable Velocity (DP IB Maths: AI HL)

Revision Note

How are vectors used to model linear motion?

How are vectors used to model motion with variable velocity?

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1. Number & Algebra

1.1 Number Toolkit

1.1.1 Standard Form

1.1.2 Approximation & Estimation

1.1.3 GDC: Solving Equations

1.2 Exponentials & Logs

1.2.1 Exponents

1.2.2 Logarithms

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.4 Financial Applications

1.4.1 Compound Interest & Depreciation

1.4.2 Amortisation & Annuities

1.5 Complex Numbers

1.5.1 Intro to Complex Numbers

1.5.2 Modulus & Argument

1.5.3 Introduction to Argand Diagrams

1.6 Further Complex Numbers

1.6.1 Geometry of Complex Numbers

1.6.2 Forms of Complex Numbers

1.6.3 Applications of Complex Numbers

1.7 Matrices

1.7.1 Introduction to Matrices

1.7.2 Operations with Matrices

1.7.3 Determinants & Inverses

1.7.4 Solving Systems of Linear Equations with Matrices

1.8 Eigenvalues & Eigenvectors

1.8.1 Eigenvalues & Eigenvectors

1.8.2 Applications of Matrices

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Further Functions & Graphs

2.2.1 Functions

2.2.2 Graphing Functions

2.2.3 Properties of Graphs

2.3 Modelling with Functions

2.3.1 Linear Models

2.3.2 Quadratic & Cubic Models

2.3.3 Exponential Models

2.3.4 Direct & Inverse Variation

2.3.5 Sinusoidal Models

2.3.6 Strategy for Modelling Functions

2.4 Functions Toolkit

2.4.1 Composite & Inverse Functions

2.5 Transformations of Graphs

2.5.1 Translations of Graphs

2.5.2 Reflections of Graphs

2.5.3 Stretches of Graphs

2.5.4 Composite Transformations of Graphs

2.6 Further Modelling with Functions

2.6.1 Properties of Further Graphs

2.6.2 Natural Logarithmic Models

2.6.3 Logistic Models

2.6.4 Piecewise Models

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.1.1 Coordinate Geometry

3.1.2 Radian Measure

3.1.3 Arcs & Sectors

3.2 Geometry of 3D Shapes

3.2.1 3D Coordinate Geometry

3.2.2 Volume & Surface Area

3.3 Trigonometry

3.3.1 Pythagoras & Right-Angled Trigonometry

3.3.2 Non Right-Angled Trigonometry

3.3.3 Applications of Trigonometry & Pythagoras

3.4 Further Trigonometry

3.4.1 The Unit Circle

3.4.2 Simple Identities