IBMathsDPApplications & Interpretation (AI)HLRevision Notes3. Geometry & TrigonometryModelling with VectorsKinematics with Vectors

Kinematics with Vectors (DP IB Applications & Interpretation (AI)) : Revision Note

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Kinematics using Vectors

  • Kinematics is the use of mathematics to model motion in objects

  • If an object is moving in one dimension then its velocity, displacement and time are related using the formula s = vt

    • where s is displacement, v is velocity and t is the time taken

  • If an object is moving in more than one dimension then vectors are needed to represent its velocity and displacement

    • Whilst time is a scalar quantity, displacement and velocity are both vector quantities

  • Vectors are often used in questions in the context of forces, acceleration or velocity

  • The position of an object at a particular time can be modelled using a vector equation

 

How do I find the direction of a vector?

  • Vectors have opposite directions if they are the same size but opposite signs

  • The direction of a vector is what makes it more than just a scalar

    • E.g. two objects with velocities of 7 m/s and ‑7 m/s are travelling at the same speed but in opposite directions

  • Two vectors are parallel if and only if one is a scalar multiple of the other

  • For real-life contexts such as mechanics, direction can be calculated from a given vector using trigonometry 

    • Given the i and j components a right-triangle can be created and the angle found using SOHCAHTOA

  • It is usually given as a bearing or as an angle calculated anticlockwise from the positive x-axis

How do I find the distance between two moving objects?

  • If two objects are moving with constant velocity in non-parallel directions the distance between them will change

  • The distance between them can be found by finding the magnitude of their position vectors at any point in time

  • The shortest distance between the two objects at a particular time can be found by finding the value of the time at which the magnitude is at its minimum value

    • Let the time when the objects are at the shortest distance be t

    • Find the distance, d, in terms of t by substituting into the equation for the magnitude of their position vectors

    • d2 will be an expression in terms of t which can be differentiated and set to 0

    • Solving this will give the time at which the distance is at a minimum

    • Substitute this back into the expression for d to find the shortest distance

Examiner Tips and Tricks

  • Kinematics questions can have a lot of information in, read them carefully and pick out the parts that are essential to the question

  • Look out for where variables used are the same and/or different within vector equations, you will need to use different techniques to find these

Worked Example

Two objects, A and B, are moving so that their position relative to a fixed point, O at time t, in minutes can be defined by the position vectors begin mathsize 16px style bold italic r subscript bold italic A blank equals blank open parentheses fraction numerator 3 over denominator negative 1 end fraction close parentheses plus t open parentheses fraction numerator negative 2 over denominator 4 end fraction close parentheses end style and begin mathsize 16px style bold italic r subscript bold italic B blank equals blank open parentheses 2 over 5 close parentheses plus t open parentheses fraction numerator 3 over denominator negative 1 end fraction close parentheses end style.

The unit vectors i and j are a displacement of 1 metre due East and North of O respectively.

a) Find the coordinates of the initial position of the two objects.

3-9-1-ib-ai-hl-kin-vectors-we-soltuion-a

b) Find the shortest distance between the two objects and the time at which this will occur.

3-9-1-ib-ai-hl-kin-vectors-we-soltuion-b
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Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

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