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Newton's Second Law of Motion
Classical Mechanics
Newton's second law is the workhorse of mechanics: it connects the force on an object to how its motion changes. It explains why a small push can send a light ball flying but barely moves a heavy crate, and why the same engine accelerates a small car faster than a loaded truck. If you can use F = ma confidently, you can solve a huge range of physics problems.
What the law says
Newton's second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In symbols, **F = m a**, where F is the net force in newtons (N), m is the mass in kilograms (kg), and a is the acceleration in metres per second squared (m/s²). Acceleration is in the same direction as the net force. Double the force and you double the acceleration; double the mass and you halve the acceleration for the same force.
The formula
The equation F = ma can be rearranged to suit the problem. To find acceleration, use a = F / m. To find mass, use m = F / a. Always use the **net** force, not just one of the forces acting: if friction opposes a push, subtract it first. One newton is defined as the force that gives a 1 kg mass an acceleration of 1 m/s², so 1 N = 1 kg·m/s². Keeping units consistent, kilograms, metres, seconds and newtons, is essential, and converting grams to kilograms or kilometres per hour to metres per second before substituting avoids most errors.
Where you see it
You see the second law everywhere. A footballer kicks a light ball hard and it accelerates sharply, while the same kick on a medicine ball produces far less acceleration. A car with a powerful engine and low mass accelerates quickly; adding passengers increases the mass and slows the acceleration for the same engine force. Rockets are an extreme case: they burn fuel so fast that their mass falls while thrust stays high, so acceleration grows as they climb. Even braking is the second law in reverse, a large braking force on a small mass gives a large deceleration.
Worked example: pushing a shopping cart
A shopper pushes a shopping cart of mass 20 kg with a net force of 50 N. Use F = m a, rearranged as a = F / m. Substitute: a = 50 N / 20 kg. a = 2.5 m/s². Answer: the cart accelerates at 2.5 m/s² in the direction of the push.
Common mistakes
- Using just one force instead of the net force. Correction: subtract opposing forces such as friction first, then use the total that remains.
- Mixing up mass and weight. Correction: mass (kg) goes into F = ma; weight is a force (mg) and is measured in newtons.
- Forgetting to convert units. Correction: convert grams to kilograms and km/h to m/s before substituting into the equation.
Practice questions
- A net force of 30 N acts on a 6 kg box. What is its acceleration?
Show answer
a = F / m = 30 / 6 = 5 m/s².
- What net force is needed to accelerate a 2 kg ball at 4 m/s²?
Show answer
F = m a = 2 x 4 = 8 N.
- A 1000 kg car accelerates at 3 m/s². What is the net force on it?
Show answer
F = m a = 1000 x 3 = 3000 N.
Frequently asked questions
Is F = ma the whole of Newton's second law?
For most school problems, yes. In full form it says force equals the rate of change of momentum, which reduces to F = ma when mass is constant.
What direction is the acceleration?
Acceleration is always in the direction of the net force, not necessarily in the direction the object is already moving.