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Velocity vectors and position Cambridge IGCSE Additional Mathematics revision

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In plain words

A velocity vector says how far something moves each second, across and up. Multiply it by the time and you have the displacement. Add that to where it started and you know where it is.

5 things to know

  1. A velocity vector gives the change in position per unit of time. Speed is the magnitude of the velocity.
  2. Position at time t = starting position + velocity × t.
  3. When two velocities act together, such as a boat's own velocity and a current, the resultant velocity is their sum.
  4. Two objects collide, or meet, if they are at the same position at the same time. Write both positions in terms of t, put the i components equal to find t, then check that the j components agree for that t.
  5. A direction of travel can be given as a unit vector. Velocity = speed × unit vector in the direction of motion.

Worked example

A particle starts at the point with position vector 2i + 3j metres and moves with velocity 4i − j metres per second. Find its position after 5 seconds, and its speed.

  1. Position = (2i + 3j) + 5(4i − j) = 22i − 2j metres.
  2. Speed = √(4² + (−1)²) = √17 = 4.12 metres per second.

Worked example

P starts at the origin with velocity 3i + 4j. Q starts at 10i + 5j with velocity i + 3j. Show that they collide, and find where.

  1. Position of P at time t: 3t i + 4t j. Position of Q: (10 + t)i + (5 + 3t)j.
  2. The i components are equal when 3t = 10 + t, so t = 5.
  3. At t = 5 the j components are 20 for P and 5 + 15 = 20 for Q. They are equal, so the particles collide.
  4. The collision is at 15i + 20j.

Tips and tricks

  • Velocity is a vector. Speed is a number. If the question asks for speed, find the magnitude.
  • For a collision, both components must match at the same time. Equal i components alone are not enough.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Velocity vectors and position: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. A particle has velocity 6i + 8j metres per second. What is its speed?
    • 14 m/s
    • 10 m/s (the answer)
    • 48 m/s
    • 2 m/s

    The magnitude: √(36 + 64).

  2. A particle starts at the origin with velocity 2i + j. What is its position at t = 3?
    • 2i + j
    • 5i + 4j
    • 6i + 3j (the answer)
    • 3i + 3j

    3 × (2i + j).

  3. A boat's velocity is 4i and the current is 3j. What is the resultant velocity?
    • 7
    • 4i + 3j (the answer)
    • 12
    • i + j

    Add the two vectors.

  4. When do two moving particles collide?
    • when they have the same velocity
    • when they have the same speed
    • when they are at the same position at the same time (the answer)
    • when their paths cross

    Paths can cross without the particles being there together.

  5. A particle moves at speed 10 in the direction of the unit vector 0.6i + 0.8j. What is its velocity?
    • 0.6i + 0.8j
    • 6i + 8j (the answer)
    • 10i + 10j
    • 16i + 18j

    Speed × unit vector.

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