Newton's second law with a variable force

Choosing the form

• Force depends on t: m dv/dt = F(t). Integrate with respect to t.
• Force depends on x: m v dv/dx = F(x). Integrate with respect to x.
• Force depends on v (for example a resistance kv): m dv/dt = -kv, then separate the variables.

Signs

Choose a positive direction and stick to it. A force towards O acting on a particle at positive x is negative. A resistance always opposes the motion.

Work done by a variable force

Work done = ∫F dx between the two positions, and it equals the change in kinetic energy. This is the same result as integrating m v dv/dx = F(x).

Worked example

A particle of mass 0.5 kg moves in a straight line under a force (3t² - 2) N. It is at rest when t = 0. Find its speed when t = 2.

1. a = F/m = 6t² - 4.
2. v = 2t³ - 4t + c, and v = 0 at t = 0 so c = 0.
3. v(2) = 16 - 8.

Answer: 8 m/s

Check yourself

1. In the worked example, find the speed when t = 3.
   Show answer

   42 m/s. v = 2(27) - 4(3) = 42 m/s.

2. A particle of mass 3 kg starts from rest at x = 0 and moves along the x-axis under a force 12/(x + 1)² N in the positive direction. Find its speed when x = 3, to 3 significant figures.
   Show answer

   2.45 m/s. 3v dv/dx = 12(x + 1)^-2, so 1.5v² = -12/(x + 1) + 12. At x = 3, 1.5v² = 9, v² = 6, v = 2.45 m/s.

3. For the same particle, find the speed it approaches as x becomes very large, to 3 significant figures.
   Show answer

   2.83 m/s. 1.5v² tends to 12, so v tends to sqrt 8 = 2.83 m/s.

4. A boat of mass 2 kg moving at 10 m/s is slowed by a resistance of 4v N, with no other horizontal force. Find the time for its speed to halve, to 3 significant figures.
   Show answer

   0.347 s. 2 dv/dt = -4v, so dv/v = -2 dt and v = 10e^-2t. v = 5 when e^-2t = ½, t = ½ ln 2 = 0.347 s.

5. A force of 3x² N moves a particle along the x-axis from x = 1 to x = 2 (distances in metres). Find the work done.
   Show answer

   7 J. ∫ from 1 to 2 of 3x² dx = 8 - 1 = 7 J.

6. A particle of mass 4 kg starts from rest and is acted on by a force of 8e^-t N. Find the speed it approaches as t becomes large.
   Show answer

   2 m/s. a = 2e^-t, v = 2(1 - e^-t), which tends to 2 m/s.

7. The force on a particle depends on its displacement x. Which equation should you write?
   • m dv/dt = F(x), integrated with respect to t
   • F(x) = m v²/x
   • m dx/dt = F(x)
   • m v dv/dx = F(x)
   Show answer

   m v dv/dx = F(x). Use v dv/dx as the acceleration so both sides involve only v and x.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M3
• Pearson Edexcel IAL Mechanics M3 (WME03) question paper, June 2025: front-cover instruction on g and accuracy

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