Newton’s Second Law Calculator relates net force to mass and a velocity change over time. It also solves F = ma for acceleration, mass, final velocity or the time a force must act.
Use a minus sign for a velocity, force or acceleration that points backward.
Result Unit and Motion Over Time
Step-by-Step Working
- Known values: m = 3,500 lb, vi = 0 mph, vf = 60 mph, Δt = 20 s
- Convert to SI units: m = 1,587.57 kg, vi = 0 m/s, vf = 26.8224 m/s
- Acceleration: a = (vf − vi) ÷ Δt = 1.34112 m/s²
- Formula: F = m × (vf − vi) ÷ Δt
- Substitute: 1,587.57 kg × (26.8224 − 0) m/s ÷ 20 s
- Result: F = 2,129.13 N
Motion While the Force Acts
| Time (s) | Velocity (m/s) | Displacement (m) | Momentum (kg·m/s) |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 5 | 6.706 | 16.76 | 10,650 |
| 10 | 13.41 | 67.06 | 21,290 |
| 15 | 20.12 | 150.9 | 31,940 |
| 20 | 26.82 | 268.2 | 42,580 |
The Law Behind the Newton’s Second Law Calculator
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. The acceleration points the same way as the force. OpenStax University Physics, section 5.3, gives the law in these words.
In simple terms, a harder push makes something speed up faster, and a heavier thing speeds up less for the same push. A child can feel it with a shopping cart. An empty cart shoots forward with one shove, and a full one barely moves. Double the force and the acceleration doubles. Double the mass and it halves.
The law is often called the law of acceleration, or the law of force and acceleration, because it is the one that says how much acceleration a force produces. Newton published it in 1687, and it is still the starting point for the other Dynamics and Statics calculators.
The Newton’s Second Law Calculator turns the statement into numbers through six Solve For modes. It finds force from a velocity change or from an acceleration, and it also solves for acceleration, mass, final velocity or the time a force must act. A minus sign marks any velocity, force or acceleration that points backward.
Newton’s Second Law Formula
The formula for the second law is force equals mass times acceleration. F is the net force in newtons, m is the mass in kilograms and a is the acceleration in meters per second squared.
$$F = m \times a$$
To calculate with Newton’s second law, pick the unknown and rearrange. Dividing by the mass gives the acceleration, and dividing by the acceleration gives the mass. The Newton’s Second Law Calculator does the rearranging when you change Solve For.
$$a = \frac{F}{m}$$
$$m = \frac{F}{a}$$
Used as an F = ma calculator, the Force from acceleration mode simply multiplies mass by acceleration. A 10 kg mass at 2 m/s² needs 20 N. In the Acceleration mode, 10,000 N on 2,000 kg gives 5 m/s². In the Mass mode, 10,000 N producing 5 m/s² means a mass of 2,000 kg.
How is 1 newton calculated?
One newton is what the formula returns for 1 kg accelerating at 1 m/s², so 1 N equals 1 kg·m/s². US units follow the same rule. A mass of 1 lb at 1 ft/s² gives one poundal, which the NIST SP 811 tables list as 0.138255 N. One slug at 1 ft/s² gives one pound-force, or 4.448222 N.
A newton is therefore not a kilogram, and 1 kg of force is not 1 newton either. One kilogram-force is the weight of 1 kg, which is 9.80665 N. The kg to Newtons Calculator handles that conversion in both directions.
The momentum form used by NCERT
Newton wrote the law in terms of momentum, the product of mass and velocity. The NCERT Class 11 textbook keeps that wording. It says the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts.
$$F = \frac{\Delta p}{\Delta t} = \frac{m (v_f – v_i)}{\Delta t}$$
When the mass stays constant, the change in velocity over time is the acceleration, and the expression becomes F = ma again. The two versions are the same law, and this second one is the form the default mode uses.
Force Without Acceleration
Many problems never state the acceleration. To calculate force without acceleration, use the mass, the starting and final velocity and the time taken. The Newton’s Second Law Calculator opens in this mode, and the result is the average force over the interval. For a 1,500 kg car going from rest to 27 m/s in 9 s, the working runs like this.
- Known values are m = 1,500 kg, vi = 0 m/s, vf = 27 m/s and Δt = 9 s.
- The acceleration is (27 − 0) ÷ 9 = 3 m/s².
- The formula is F = m × (vf − vi) ÷ Δt, which gives 1,500 × 27 ÷ 9.
- The result is a net force of 4,500 N in the direction of travel.
The cards restate the answer as 1,012 lbf, 458.9 kgf and 32,550 poundals. They also show the acceleration as 9.843 ft/s² or 0.3059 g, and a distance of 121.5 m covered while the force acts.
F is always the net force. The engine in this example has to supply more than 4,500 N, because drag and rolling resistance push the other way. When the acceleration has to come from a distance instead of a time, the Acceleration Calculator finds it first. Forces from pressure or a spring are covered by the Force Calculator.
Impact force and average force
The same mode turns the Newton’s Second Law Calculator into an average force calculator for stops. A 3,500 lb car braking from 60 mph to rest in 20 s needs −2,129 N, which is −478.6 lbf. The minus sign shows that the force points against the motion.
Shorten the time and the force climbs in proportion. If the 1,500 kg car stops from 27 m/s in 0.15 s, the net force is −270,000 N, or −270 kN. That is 60,700 lbf and a deceleration of 18.35 g. The momentum change matches the 9 s case, but the time is 60 times shorter, so the force is 60 times larger.
That is the whole idea behind an impact force estimate. The Car Crash Calculator takes it further and works out the load on a belted or unbelted occupant from the stopping distance.
Newton-Seconds, Work and Power
To calculate newton seconds, multiply the net force by the time it acts. The product is called impulse, and it equals the change in momentum. The Newton’s Second Law Calculator reports it on the fourth card.
$$J = F \times \Delta t = m \times \Delta v$$
For the accelerating car, 4,500 N over 9 s is 40,500 N·s, the same as 1,500 kg times 27 m/s. The card lists it alongside the velocity change and the average velocity, both converted to a second speed unit.
The third card follows the motion. Displacement is the average velocity times the time, here 121.5 m. Work is force times displacement, 546,750 J, which is exactly the kinetic energy the car gains. Dividing by the time gives an average power of 60,750 W.
The Motion While the Force Acts table splits the interval into quarters and lists velocity, displacement and momentum at each step. The Final velocity and Time needed modes run the same equation backward. A force of 4,500 N on 1,500 kg for 9 s returns 27 m/s, and asking for the time to reach 27 m/s returns 9 s.
How Much Force Is 2, 10, 30 or 400 Newtons?
A force in newtons is easier to picture through what it does. The table shows the acceleration each force gives a 1 kg mass and a 70 kg person, and the mass whose weight on Earth equals that force.
| Net force | Acceleration of 1 kg | Acceleration of 70 kg | Equal to the weight of | In pounds-force |
|---|---|---|---|---|
| 1 N | 1 m/s² | 0.01429 m/s² | 0.102 kg | 0.2248 lbf |
| 2 N | 2 m/s² | 0.02857 m/s² | 0.2039 kg | 0.4496 lbf |
| 10 N | 10 m/s² | 0.1429 m/s² | 1.02 kg | 2.248 lbf |
| 30 N | 30 m/s² | 0.4286 m/s² | 3.059 kg | 6.744 lbf |
| 400 N | 400 m/s² | 5.714 m/s² | 40.79 kg | 89.92 lbf |
Ten newtons is just over the weight of a 1 kg bag, and 30 N is about the weight of 3 kg. A net force of 400 N would speed up a 70 kg person at 5.714 m/s², more than half of g. Enter any of these in the Newton’s Second Law Calculator with your own mass to see the acceleration it would cause.
Newton’s First, Second and Third Laws
The three laws each have a short formula. The first law says that with zero net force the velocity does not change, so a = 0. The second law is F = ma. The third law says that when body A pushes on body B, B pushes back on A with an equal and opposite force.
$$F_{AB} = -F_{BA}$$
The Newton’s Second Law Calculator shows the first law directly. Enter a net force of zero, or equal starting and final velocities, and it reports Net Force Is Zero with the velocity unchanged.
Textbooks often call the second law the real law of motion. It is the only one of the three that gives a number, and the first law is simply its special case with F set to zero.
How to prove Newton’s second law
The law is proved by experiment, not by algebra. A cart on a low-friction track is pulled by a falling weight while a timer records its acceleration. Doubling the pull with the mass fixed doubles the acceleration. Doubling the mass with the pull fixed halves it. The Acceleration mode gives the predicted value to compare with each run.
Is 9.8 m/s² a Velocity?
It is an acceleration. The value 9.8 m/s² means a falling object gains 9.8 m/s of speed every second, so gravity has no single speed. After one second of free fall the object moves at 9.8 m/s, and after two seconds at 19.6 m/s. The acceleration field of the Newton’s Second Law Calculator accepts g as a unit, and one g is 9.80665 m/s².
The figure comes from the second law combined with Newton’s law of gravitation. Setting the weight m × g equal to the gravitational pull lets the mass cancel, and what remains depends only on Earth.
$$g = \frac{G M}{R^2}$$
With Earth’s mass and mean radius this gives about 9.82 m/s², which the Gravitational Force Calculator works through step by step. Physics problems round it to 9.8 or 9.81, and 10 m/s² is a mental shortcut that runs 2% high.