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

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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)
      • lambda 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 bold italic p equals bold italic a plus t bold italic b

Find the vectors a and b.

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

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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 equals blank fraction numerator straight d s over denominator straight d t end fraction
  • In more than one dimension vectors are used to represent motion
  • For displacement given as a function of time in the form
    • bold r open parentheses t close parentheses equals open parentheses fraction numerator f subscript 1 open parentheses t close parentheses over denominator f subscript 2 open parentheses t close parentheses end fraction close parentheses
  • The velocity vector can be found by differentiating each component of the vector individually
    • begin mathsize 16px style bold v equals blank open parentheses fraction numerator v subscript 1 open parentheses t close parentheses over denominator v subscript 2 open parentheses t close parentheses end fraction close parentheses end style
    • begin mathsize 16px style bold v equals blank fraction numerator d bold r over denominator d t end fraction equals blank open parentheses fraction numerator f subscript 1 apostrophe open parentheses t close parentheses over denominator f subscript 2 apostrophe open parentheses t close parentheses end fraction close parentheses end style
    • 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
    • begin mathsize 16px style bold a equals blank fraction numerator d bold v over denominator d t end fraction equals blank fraction numerator d squared bold r over denominator d t squared end fraction end style
  • In two dimensions acceleration can be found by differentiating each component of the velocity vector individually
    • begin mathsize 16px style bold a equals open parentheses fraction numerator a subscript 1 open parentheses t close parentheses over denominator a subscript 2 open parentheses t close parentheses end fraction close parentheses end style
    • begin mathsize 16px style bold a equals blank fraction numerator d bold v over denominator d t end fraction equals open parentheses fraction numerator v subscript 1 apostrophe open parentheses t close parentheses over denominator v subscript 2 apostrophe open parentheses t close parentheses end fraction close parentheses end style
    • begin mathsize 16px style bold a equals fraction numerator d squared bold r over denominator d t squared end fraction equals blank open parentheses fraction numerator f subscript 1 apostrophe apostrophe open parentheses t close parentheses over denominator f subscript 2 apostrophe apostrophe open parentheses t close parentheses end fraction close parentheses end style
  • 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 space v equals space open parentheses 5
3 close parentheses plus t open parentheses space space space space 0
minus 0.8 close parentheses. At the time t = 0 the position vector of the ball is 3i-2j.

a)
Find the acceleration vector of the ball's motion.

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

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

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

 

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Amber

Author: Amber

Expertise: Maths

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.