Force, mass and acceleration: the link with the second law

A reminder of Newton’s second law

To use the third law effectively, it must be linked to Newton’s second law, which relates force and acceleration:

F = m × a

where F is the net force (in newtons, N), m is the mass (in kg), and a is the acceleration (in m/s²).

Same force, different effects

A key point: in an action-reaction pair, the two forces are equal in magnitude, but the accelerations produced can be very different if the masses are different.

Example: a skater with a mass of 80 kg pushes against a wall. The wall exerts a reaction force F on the skater. The skater’s acceleration is a = F / 80. If, instead of a wall, he pushes another skater of mass 40 kg on skates, the latter will experience the same force F, but his acceleration will be a’ = F / 40, which is twice that of the first skater.

Summary table

System	Force received	Mass	Acceleration
Skater 1 (80 kg)	F	80 kg	F/80
Skater 2 (40 kg)	F (same value)	40 kg	F/40 = 2 × (F/80)

Pitfall to avoid

Do not confuse ‘same force’ with ‘same effect’. Newton’s third law guarantees that forces are equal, but by no means that accelerations or resultant velocities are equal: these depend on the mass of each object via F = m × a.

The case of collisions

During a collision between two cars of very different masses, the force experienced by each vehicle is the same (in absolute terms), in accordance with the third law. However, the lighter car will experience a much greater acceleration (and therefore more damage and deceleration) than the heavier car, because a = F/m.