Integrating velocity and acceleration vectors

Integrating

• v = ∫ a dt, r = ∫ v dt, each component separately.
• The constant is a vector c = pi + qj. If a particle is at 2i + j when t = 0, then r(0) = 2i + j fixes c.

From force to motion

If the force is given as a function of t, divide by the mass to get a, then integrate. A particle that "starts from rest at O" has v = 0 and r = 0 when t = 0.

Typical follow-up parts

Find the speed at a given time, when the particle is moving due north or due east, or its distance from O. Keep the vector form until the last step, then take the magnitude.

Worked example

A particle has acceleration a = 6t i - 4j m/s². At t = 0 it is at O with velocity (2i + 3j) m/s. Find its position vector when t = 2.

1. v = 3t² i - 4t j + c, and v(0) = 2i + 3j, so v = (3t² + 2)i + (3 - 4t)j.
2. r = (t³ + 2t)i + (3t - 2t²)j + 0 (it starts at O).
3. At t = 2: r = (8 + 4)i + (6 - 8)j = 12i - 2j.

Answer: r = (12i - 2j) m

Check yourself

1. A particle starts at O with v = 3t² i + 2t j m/s. Find its distance from O when t = 2, to 3 significant figures.
   Show answer

   8.94 m. r = t³ i + t² j, so r(2) = 8i + 4j and the distance is sqrt(80) = 8.94 m.

2. A particle has a = 2i + 6t j m/s² and velocity (4i - 3j) m/s when t = 0. Find its speed when t = 1.
   Show answer

   6 m/s. v = (4 + 2t)i + (-3 + 3t²)j; at t = 1, v = 6i + 0j, so the speed is 6 m/s.

3. A particle has velocity v = (2t - 4)i + 3j m/s, with j due north. Find the time when it is moving due north.
   Show answer

   2 s. Due north: i part zero, 2t - 4 = 0, so t = 2 (and the j part 3 is positive).

4. A particle of mass 2 kg starts from rest and is acted on by a force F = 4t i + 6j N. Find its speed when t = 3, to 3 significant figures.
   Show answer

   12.7 m/s. a = F/2 = 2t i + 3j, so v = t² i + 3t j. At t = 3, v = 9i + 9j and speed = 12.7 m/s.

5. A particle has v = 4t i + 3j m/s and is at (2i + j) m when t = 0. Find its distance from O when t = 2, to 3 significant figures.
   Show answer

   12.2 m. r = (2t² + 2)i + (3t + 1)j, so r(2) = 10i + 7j and the distance is sqrt(149) = 12.2 m.

6. When you integrate a velocity vector to get a position vector, what is the constant of integration?
   • A single number added to the i part only
   • A vector, such as pi + qj
   • Always the zero vector
   • The acceleration at t = 0
   Show answer

   A vector, such as pi + qj. Each component gets its own constant, so together they form a vector.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M2
• Pearson Edexcel IAL Mechanics M2 (WME02) question paper, June 2025: front-cover instructions on g and accuracy

Last updated 8 November 2026. Spotted a mistake? Tell us.