The Second Law: F = ma
Force sets acceleration, and mass decides how much.
Lesson at a glanceShow the key points
- Second law: in an inertial frame, a = F/m, or F = ma, with F the vector sum of all forces on the particle.
- It is instantaneous: when the net force vanishes, the acceleration vanishes at that same instant.
- It is a vector equation: Fₓ = maₓ, F_y = ma_y, F_z = ma_z apply separately.
- 1 N is the force that gives a 1 kg mass an acceleration of 1 m/s².
- Combine F = ma with the equations of motion to link forces to stopping distances, times and speeds.
The first law says what happens when the forces cancel. The second says exactly what happens when they don’t.
The second law
Measured from an inertial frame, the acceleration of a particle equals the vector sum of all the forces on it divided by its mass:
Here is the net force (the resultant of every force acting on the particle) and the acceleration points the same way as that net force. The first law is the special case . This is a law of nature: once inertial frames are defined, it tells us how any force changes any motion.
A vector equation
holds component by component, which is how we use it in practice:
This also gives the newton its precise meaning: 1 N is the net force that gives a 1 kg mass an acceleration of 1 m/s², so 1 N = 1 kg·m/s².
A 25 g arrow travelling at 60 m/s sinks 8.0 cm into a straw target before stopping. Assuming the target’s resisting force is constant, how large is it?
A 2.0 kg puck, initially at rest on frictionless ice at the origin, is pushed by a constant force . Where is it, and how fast is it going, after 4.0 s?
- 1 with the net force, measured in an inertial frame.
- 2It is instantaneous: no force, no acceleration, at that very moment.
- 3Apply it component by component; 1 N = 1 kg·m/s².
- 4Combine with kinematics to connect forces with distances, times and speeds.
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