Newton’s second law: force, mass and acceleration (F = ma)

Newton's second law says that the net force on an object equals its mass times its acceleration: F = ma. Push twice as hard and an object accelerates twice as fast; push something twice as massive with the same force and it accelerates half as fast. It is the workhorse equation of mechanics, and the force calculator solves it for force, mass or acceleration in whatever units you have.

What the law says

Isaac Newton set out his three laws of motion in the Principia, published in 1687. The first says an object stays still, or keeps moving at a steady velocity, unless a net force acts on it. The second says what happens when one does: the object accelerates in the direction of the net force, in proportion to the force and in inverse proportion to its mass. The third says forces come in pairs: push on a wall and the wall pushes back on you just as hard. In its more general form the second law says that force equals the rate of change of momentum; F = ma is the version for an object whose mass does not change.

The word that matters most is net. F is the sum of all the forces acting, with their directions taken into account. A car cruising at a steady 100 km/h has a large driving force from its engine, but it is balanced by air resistance and rolling resistance, so the net force and the acceleration are both zero.

Newtons, kilograms and m/s²

In SI units, mass is measured in kilograms, acceleration in metres per second squared and force in newtons. The newton is defined by the law itself: 1 N is the force that gives 1 kg an acceleration of 1 m/s², so 1 N = 1 kg·m/s². That is roughly the weight of a 100 g apple. Other units turn up in engineering and older textbooks: one pound-force (lbf) is 4.448 N, one kilogram-force (kgf) is 9.80665 N, and the dyne, from the centimetre-gram-second system, is 0.00001 N. Mixing units is the commonest source of wrong answers, which is why the calculator converts every input first; Metric vs imperial: how we got two systems explains how the two systems relate.

Weight is the second law in disguise. Near Earth's surface every object in free fall accelerates at about 9.81 m/s² (standard gravity is defined as exactly 9.80665 m/s²), so the gravitational force on a mass m is W = mg. A 70 kg person weighs about 687 N. Their mass would be the same on the Moon, but their weight would not, as the weight on other planets calculator shows.

Worked examples

• Accelerating a car. A 1,500 kg car accelerating at 3 m/s² needs a net force of 1,500 × 3 = 4,500 N, about 1,012 lbf. The same force on a 3,000 kg van gives only 1.5 m/s².
• Pushing a trolley. You push a 20 kg trolley with 50 N and friction resists with 10 N. The net force is 40 N, so a = 40 ÷ 20 = 2 m/s².
• A lift setting off upwards. An 800 kg lift car accelerating upwards at 1.5 m/s² needs a cable tension of m(g + a) = 800 × (9.81 + 1.5) ≈ 9,050 N, compared with about 7,850 N when it hangs still or moves at a steady speed. The extra force is why you feel heavier as a lift sets off upwards.
• Braking. Stopping a 1,500 kg car from 20 m/s (72 km/h) in 4 seconds is an average deceleration of 5 m/s², which needs an average braking force of 7,500 N.

Forces are vectors

Force has a direction as well as a size, so forces at angles cannot simply be added together. Each one is split into horizontal and vertical components (F cos θ and F sin θ), the components are summed separately, and Pythagoras recombines them. Two 10 N forces at right angles give a resultant of √(10² + 10²) ≈ 14.14 N at 45°. The force calculator's second panel does this for any number of forces, and if the resultant comes out as zero, the forces are balanced and the object does not accelerate. On a slope the same idea splits gravity into one component along the slope and one pressing into it, which is what the inclined plane calculator works with.

Momentum and falling objects

Writing the law as force equals change in momentum divided by time, F = Δp ÷ Δt, explains a lot of safety engineering. For the same change in momentum, stretching out the stopping time reduces the force: crumple zones, airbags and bending your knees as you land all work this way. The momentum calculator covers impulse and collisions.

The law also explains a famous result. The gravitational force on an object is proportional to its mass, so the acceleration it produces, F ÷ m, is the same for every object. Without air resistance a hammer and a feather fall together, as an Apollo 15 astronaut showed on the Moon. On Earth, air resistance grows with speed until it balances weight and the object stops accelerating; the free fall calculator includes that terminal velocity, and Escape velocity explained: how fast you need to go to leave a planet follows gravity much further out.