In short
Newton's second law says that the net force on an object equals the rate of change of its momentum: \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}. For an object whose mass stays constant, this simplifies to \vec{F}_{\text{net}} = m\vec{a} — the net force equals mass times acceleration. Force is measured in newtons (1 N = 1 kg·m/s²). Weight is a special case: W = mg, where g \approx 9.8 m/s² near Earth's surface.
You are standing behind a heavy steel almirah that needs to go across the room. You push. Nothing happens. You push harder — your shoes slip on the floor, your arms shake, but the almirah barely budges. Then your younger sibling, half your mass, pushes a plastic chair with one hand and the chair shoots across the room.
Two pushes. Two very different results. The almirah is massive and needs an enormous force to get moving. The chair is light and responds to a small push with a large acceleration. You already sense the pattern: how much something accelerates depends on how hard you push and how heavy it is. Newton's second law makes this intuition precise — and it turns out to be the single most useful equation in all of physics.
The idea: force changes motion
Newton's first law told you that an object left alone keeps doing what it was already doing — moving at constant velocity or sitting still. The second law answers the next question: what happens when the object is not left alone? When something pushes or pulls on it, how does its motion change?
The answer is built on a quantity you already know: momentum. The momentum of an object is its mass times its velocity:
Momentum captures how much "motion" an object carries. A 60 kg sprinter running at 10 m/s has 600 kg·m/s of momentum. A 1200 kg car crawling at 0.5 m/s also has 600 kg·m/s of momentum. They carry the same amount of motion — it would be equally hard to stop either one.
Newton's deep insight was this: force is what changes momentum. Not velocity. Not position. Momentum. The net force on an object equals the rate at which its momentum changes with time.
The law in its full form: \vec{F} = \frac{d\vec{p}}{dt}
Newton stated the second law in terms of momentum (which he called "quantity of motion"). In modern notation:
This is the most general form of the second law. It says: apply a net force to something, and its momentum changes. The bigger the force, the faster the momentum changes. The direction of the force is the direction in which momentum changes.
Why momentum and not velocity? Because momentum accounts for mass. A force acting on a heavy object produces less velocity change than the same force acting on a light object — but the momentum change depends only on the force and the time, not on the mass. This makes F = dp/dt more fundamental than F = ma.
Now comes the critical step: deriving the form you will use most often.
Deriving F = ma from F = dp/dt
For the vast majority of physics problems you will encounter — a car accelerating, a ball being thrown, a block sliding down a ramp — the mass of the object does not change during the motion. The car does not shed parts as it speeds up. The ball does not gain mass mid-flight. When mass is constant, the second law simplifies beautifully.
Step 1. Start with the momentum form of the second law.
Why: this is Newton's second law in its most general form — the foundation from which everything else follows.
Step 2. Substitute the definition of momentum, \vec{p} = m\vec{v}.
Why: momentum is mass times velocity. Replacing \vec{p} with m\vec{v} makes the mass explicit so you can see what happens when it is constant.
Step 3. Apply the derivative. Since the mass m is constant, it comes out of the derivative.
Why: the derivative of a constant times a function equals the constant times the derivative of the function. This step is only valid when m does not change with time. For a rocket burning fuel, mass changes and you cannot pull m out — you need the full product rule, which gives you the rocket equation.
Step 4. Recognise that \frac{d\vec{v}}{dt} is the definition of acceleration, \vec{a}.
Why: acceleration is the rate of change of velocity. So F = dp/dt becomes F = m \cdot dv/dt = ma. Four symbols. The most used equation in physics.
Read this equation carefully. It does not say "force equals mass times acceleration." It says the net force — the vector sum of every force acting on the object — equals mass times acceleration. If three forces act on a block, you add them as vectors first, and only the resultant drives the acceleration.
What each part means
| Symbol | What it is | SI unit |
|---|---|---|
| \vec{F}_{\text{net}} | The vector sum of all forces on the object | newton (N) |
| m | The mass of the object — a measure of its inertia | kilogram (kg) |
| \vec{a} | The acceleration — how rapidly the velocity changes | m/s² |
The equation tells you three things at once:
- Direction: the acceleration points in the same direction as the net force. Push an autorickshaw northward; it accelerates northward.
- Proportionality to force: double the net force, double the acceleration (if mass is fixed).
- Inverse proportionality to mass: double the mass, halve the acceleration (if force is fixed). This is why the almirah barely moves but the plastic chair flies.
The newton — the unit of force
The SI unit of force is the newton, symbolised N. From F = ma:
One newton is the force needed to accelerate a 1 kg mass at 1 m/s². That is a gentle push — about the force you feel when you hold a small apple (mass ≈ 100 g) in your palm, since Earth's gravity pulls it down with F = 0.1 \times 9.8 \approx 1 N.
For larger forces, you will encounter kilonewtons (kN): a small car's engine produces roughly 3–5 kN of driving force; the thrust of an ISRO PSLV rocket at liftoff is about 4,430 kN.
Why is the unit named after Newton? Because this law — along with the first and third — is the foundation of classical mechanics. Every bridge, every satellite orbit, every car braking system is designed using F = ma.
How to apply the second law — a recipe
Whenever you face a problem that asks "find the force" or "find the acceleration," follow this procedure:
-
Choose your system. Decide which object (or group of objects) you are analysing. Draw a boundary around it — mentally or on paper.
-
Draw a free body diagram (FBD). Replace every contact surface, string, spring, or field with the force it exerts on your object. Label each force with a name and a direction. Show only the forces acting on your object — not the forces your object exerts on other things.
-
Choose a coordinate system. Pick positive x and positive y directions. For inclined planes, align one axis along the surface.
-
Write \vec{F}_{\text{net}} = m\vec{a} in component form. Add up all force components along each axis:
- Solve. You now have equations with numbers. Find the unknown.
This recipe works for every mechanics problem you will ever see — from a block on a table to a satellite in orbit. The physics is always in the free body diagram and the identification of forces.
Weight: gravity's version of F = ma
You stand on a bathroom scale. The scale reads a number — say, 60 kg. But what the scale actually measures is not your mass. It measures the force you exert on it. That force has a specific name: your weight.
Weight is the gravitational force that Earth exerts on you. Near Earth's surface, every kilogram of mass feels a gravitational acceleration g \approx 9.8 m/s² directed downward. Applying F = ma with a = g:
For a 60 kg person:
Why does W = mg follow from F = ma? Because gravity is a force, and the acceleration it produces (in the absence of all other forces) is g. The second law says the gravitational force on mass m is F = mg. This force is what you call weight.
Weight depends on where you are
The value of g is not a universal constant — it depends on the planet and your distance from its centre.
| Location | g (m/s²) | Weight of 60 kg person (N) |
|---|---|---|
| Earth (sea level, poles) | 9.832 | 590 |
| Earth (sea level, equator) | 9.780 | 587 |
| Delhi (≈ 220 m elevation) | 9.791 | 587 |
| Mount Everest summit | 9.773 | 586 |
| Moon surface | 1.625 | 97.5 |
| Mars surface | 3.721 | 223 |
| Jupiter surface | 24.79 | 1,487 |
Your mass is 60 kg everywhere — mass is an intrinsic property that does not change with location. But your weight changes because g changes. On the Moon, you weigh about one-sixth of what you weigh on Earth. On Jupiter, you would weigh nearly 2.5 times more.
Why the difference between mass and weight? Mass measures how much matter you contain and how hard you are to accelerate (inertia). Weight measures how strongly gravity pulls on you at a particular location. A 60 kg astronaut has 60 kg of mass whether they are in Delhi, on the Moon, or floating in the ISS — but their weight can be 588 N, 97.5 N, or effectively 0 N.
The bathroom scale in your house is calibrated to display mass in kilograms, but it actually measures force (weight). It works correctly only at Earth's surface where g \approx 9.8 m/s². Take the same scale to the Moon and it would read about 10 kg — not because your mass changed, but because the gravitational force is weaker and the scale interprets the smaller force as a smaller mass.
Free body diagram — the visual heart of the second law
Before solving any problem, you need to see all the forces. Here is the free body diagram for a book resting on a table — the simplest possible case.
The book is in equilibrium: a = 0. Applying the second law vertically:
The normal force exactly balances the weight. This is not a coincidence and not Newton's third law — it is the second law with zero acceleration.
Worked examples
Example 1: Accelerating a car
A 1200 kg car accelerates from rest to 60 km/h in 8 seconds on a straight, level road. Find the net force required.
Setting up the problem. The car starts at rest (u = 0) and reaches v = 60 km/h. Convert to SI:
Why convert? The second law works in SI units: kilograms, metres per second, newtons. Mixing km/h in would give a wrong numerical answer.
Step 1. Find the acceleration.
The car goes from 0 to 16.67 m/s in 8 s. Assuming uniform acceleration:
Why: acceleration is the rate of change of velocity. Since the car starts from rest and reaches 16.67 m/s in 8 s, the average acceleration is 16.67/8.
Step 2. Apply Newton's second law.
The net force along the direction of motion is:
Why: the second law connects net force directly to mass and acceleration. The 1200 kg mass and 2.08 m/s² acceleration multiply to give a net force of 2500 N.
Step 3. Interpret the result.
The net force is 2.5 kN directed forward. This is the net force — the difference between the engine's driving force and all resistive forces (tyre friction, air drag). If total resistance is, say, 500 N, then the engine must produce 2500 + 500 = 3000 N of driving force through the tyres.
Result: The net force required is F_{\text{net}} = 2500 N = 2.5 kN, directed forward.
What this shows: A family car needs about 2.5 kN of net force to go from rest to 60 km/h in 8 seconds. That is roughly the force needed to lift a 250 kg object against gravity — significant, but well within the capability of a typical 80–100 hp engine. Notice how doubling the car's mass (loading it with passengers and luggage) would require doubling the net force to achieve the same acceleration.
Example 2: Apparent weight in a lift
A 60 kg person stands on a bathroom scale inside a lift (elevator). The lift accelerates upward at 2 m/s². What does the scale read?
Setting up the problem. The scale does not read your mass — it reads the normal force N that you push down on it with (and, by Newton's third law, the force it pushes up on you with). When the lift accelerates, this normal force changes. That changed reading is your apparent weight.
Step 1. Identify the forces on the person.
Two forces act on the person:
- Weight, W = mg = 60 \times 9.8 = 588 N, directed downward
- Normal force, N, from the scale surface, directed upward
Why only these two? The person is touching exactly one surface (the scale) and is subject to exactly one field force (gravity). No ropes, no pushes, no springs. Two contact points, two forces.
Step 2. Apply Newton's second law in the vertical direction.
Take upward as positive. The person accelerates upward at a = 2 m/s².
Why N - mg and not mg - N? Because you chose upward as positive. The normal force N points up (positive) and the weight mg points down (negative). The acceleration is upward (positive), so ma is positive.
Step 3. Solve for N.
Why does N increase? The person is accelerating upward, so the net force must be upward. Since gravity pulls down at 588 N, the scale must push up with more than 588 N to produce an upward net force. The extra force (ma = 120 N) is what accelerates the person.
Step 4. Convert to what the scale displays.
The scale is calibrated to display mass in kilograms (it divides the measured force by g = 9.8):
Why does the scale show more than 60 kg? The scale measures force, not mass. It assumes g = 9.8 m/s² and divides. When the lift accelerates upward, the force is larger, so the scale's "mass" reading goes up — even though your actual mass has not changed.
Result: The scale reads 72.2 kg (an apparent weight of 708 N), even though the person's actual mass is 60 kg.
What this shows: When a lift accelerates upward, you feel heavier — your apparent weight increases by ma. When the lift accelerates downward, you feel lighter (the formula gives N = m(g - a), and N < mg). If the lift were in free fall (a = g downward), N = 0 — the scale reads zero and you are weightless. This is exactly the principle behind the "weightlessness" astronauts experience in orbit: the spacecraft and the astronaut are both in free fall, so the floor exerts no normal force.
Common confusions
-
"Force causes velocity." No — force causes acceleration, which is a change in velocity. An object can have a large velocity with zero net force (a spacecraft coasting through empty space). And an object can have zero velocity but a nonzero net force (a ball at the topmost point of its flight — it has v = 0 at that instant, but a = g downward because gravity is still pulling it).
-
"Heavier objects need more force to move." Almost right, but the language is sloppy. Heavier objects need more force to achieve the same acceleration. A 20 kg suitcase and a 5 kg backpack both need some force to start moving; what differs is the acceleration each force produces. If you apply 10 N to both, the backpack accelerates at 10/5 = 2 m/s² while the suitcase accelerates at 10/20 = 0.5 m/s².
-
"F = ma always applies." It applies when mass is constant. For a system whose mass changes — a rocket burning fuel, a conveyor belt loading sand, rain falling into an open cart — you need the full form F = dp/dt. These are the problems where the momentum form of the second law earns its keep.
-
"Weight and mass are the same thing." They are not. Mass is a property of the object (measured in kg); weight is a force exerted by gravity (measured in N). Your mass is the same on the Moon as on Earth; your weight on the Moon is about one-sixth of your weight on Earth. The confusion comes from bathroom scales displaying kilograms — they measure force but report mass by dividing by g.
-
"The normal force is always equal to the weight." Only when the object is on a horizontal surface with no vertical acceleration. On an incline, the normal force equals mg\cos\theta, which is less than mg. In an accelerating lift, N = m(g + a) or m(g - a) depending on the direction. The normal force is whatever value the second law requires to match the acceleration — it is not "automatically" equal to mg.
If you came here to understand what F = ma means, how to derive it from momentum, and how to use it in problems, you have everything you need. What follows is for readers preparing for JEE Advanced who want the momentum form explored further and the variable-mass case introduced.
The impulse-momentum theorem
Rearranging \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt} and integrating both sides over a time interval from t_1 to t_2:
Why: the integral of dp/dt with respect to t is just the net change in p — this is the fundamental theorem of calculus applied to momentum.
The left side is called the impulse \vec{J}:
Impulse equals the change in momentum. This is the impulse-momentum theorem — a direct consequence of Newton's second law. It is especially useful when the force acts for a short time (a bat hitting a cricket ball, a car crashing into a barrier), because you can relate the average force to the momentum change even without knowing the exact force-vs-time curve.
For a constant force over a time interval \Delta t:
A fast bowler delivers a cricket ball of mass 0.16 kg, changing its velocity from 0 to 140 km/h (38.9 m/s). The momentum change is \Delta p = 0.16 \times 38.9 = 6.22 kg·m/s. If the bowler's arm delivers the force over 0.1 s, the average force is 6.22/0.1 = 62.2 N. If the ball hits the batsman's bat and is deflected in 0.005 s, the average force during impact is 6.22/0.005 = 1244 N — twenty times larger, because the same momentum change happens in a much shorter time.
Variable mass: the rocket equation (preview)
For a system whose mass changes — a rocket expelling exhaust gases — you cannot simplify F = dp/dt to F = ma. You need the full product rule:
Why the extra term? Because both m and \vec{v} are changing with time. The product rule from calculus gives two terms: one for the velocity change of the remaining mass, one for the momentum carried away by the ejected mass.
For a rocket, the external force is gravity (-mg downward) and the thrust comes from the momentum of the exhaust. Rearranging gives the Tsiolkovsky rocket equation, which governs every launch from Sriharikota:
where v_e is the exhaust velocity, m_0 is the initial mass, and m_f is the final mass after fuel is burnt. ISRO's PSLV, for example, has a mass ratio m_0/m_f \approx 3, and exhaust velocities of 2,500–3,000 m/s for solid boosters. The derivation of this equation is covered in the variable-mass systems article.
Dimensional analysis of F = ma
Check that F = ma is dimensionally consistent:
This is not just a consistency check — it tells you something deep. Force has dimensions of mass × length / time². Any valid formula for force must have these dimensions. This means:
- Spring force F = kx: the spring constant k must have units of N/m = kg/s².
- Gravitational force F = GMm/r^2: the constant G must have units of N·m²/kg² = m³/(kg·s²).
- Drag force F = \frac{1}{2}C_D \rho A v^2: check that density × area × velocity² gives kg/m³ × m² × m²/s² = kg·m/s² = N. It does.
Dimensional analysis cannot tell you the numerical constants (like the \frac{1}{2} in the drag formula), but it constrains the form of every force law.
Where this leads next
- Newton's Third Law — every force has an equal and opposite reaction force. The companion law that completes the picture.
- Free Body Diagrams — a systematic treatment of how to identify and draw all forces on a body, including tension, friction, and normal forces.
- Applications: Connected Bodies and Systems — pulleys, Atwood machines, blocks on inclines — problems that combine the second law with constraints.
- Impulse and Momentum — the time-integrated form of the second law, and why it matters for collisions and sudden forces.
- Friction — the force that resists sliding. Static and kinetic friction, the role of the normal force, and why friction is essential for walking, driving, and braking.