Newton’s Second Law and Inertial Mass

How forces change velocity

Pushing an empty shopping trolley changes its motion more easily than pushing a heavily loaded one. Newton’s Second Law connects the three quantities involved: resultant force, mass and acceleration.

Acceleration means a change in velocity per unit time. An object accelerates when it speeds up, slows down or changes direction. Its acceleration is in the direction of the resultant force.

The resultant force is the overall effect of all the forces acting on an object, including their directions. If a trolley is pushed forwards with 10 N while resistance acts backwards with 4 N, the resultant force is 6 N forwards. It is this 6 N that determines the trolley’s acceleration.

Connecting force, mass and acceleration

Newton’s Second Law is expressed as:

$$F = ma$$

Here, $F$ is the resultant force in newtons (N), $m$ is mass in kilograms (kg), and $a$ is acceleration in metres per second squared (m/s²).

Rearranging gives two other useful forms:

$$a = \frac{F}{m} \qquad m = \frac{F}{a}$$

These allow you to calculate whichever quantity is unknown. For example, suppose a trolley has a mass of 3 kg and a resultant force of 6 N acts on it. Its acceleration is $a = 6/3 = 2\text{ m/s}^2$. This means its velocity changes by 2 m/s each second while that resultant force remains constant.

Alternatively, a 900 kg car accelerating at 2 m/s² needs a resultant force of $F = 900 \times 2 = 1800\text{ N}$. This is the overall forwards force after any opposing forces have been taken into account.

What happens when force or mass changes?

For a constant mass, acceleration is directly proportional to resultant force:

$$a \propto F$$

The symbol $\propto$ means ‘is proportional to’. Doubling the resultant force doubles the acceleration; tripling it triples the acceleration.

The graph below shows this relationship for a trolley of constant mass. The straight line through the origin represents direct proportionality: zero resultant force gives zero acceleration.

Data for Acceleration and resultant force for a constant mass

For a constant resultant force, acceleration is inversely proportional to mass:

$$a \propto \frac{1}{m}$$

Doubling the mass halves the acceleration. For example, the same resultant force of 6 N produces an acceleration of 2 m/s² in a 3 kg trolley, but only 1 m/s² in a 6 kg trolley. This explains why a loaded trolley changes velocity less readily under the same resultant force.

Showing an approximate answer

The symbol $\sim$ indicates an approximate value or answer. It is different from $\propto$: approximation means ‘about’, whereas proportionality describes a relationship between quantities.

Suppose a trolley’s mass is estimated as $m \sim 3\text{ kg}$ and its acceleration as $a \sim 2\text{ m/s}^2$. Using $F = ma$ gives $F \sim 6\text{ N}$. Because the starting values are estimates, the calculated force is also approximate.

Inertia and inertial mass — Higher Tier only

Inertia is the tendency of an object to continue in its state of rest or uniform motion. Uniform motion means constant velocity: the same speed in the same direction. A resultant force is needed to change that velocity, rather than simply to keep an object moving.

Inertial mass measures how difficult it is to change an object’s velocity. It is defined as the ratio of resultant force to acceleration:

$$m = \frac{F}{a}$$

For the same resultant force, an object with a larger inertial mass has a smaller acceleration. This resistance to a change in velocity applies to slowing down and changing direction as well as speeding up.

If the same resultant force acts on several objects, the one with the smallest acceleration has the largest inertial mass.

Newton’s Second Law

$$F = ma \qquad a = \frac{F}{m} \qquad m = \frac{F}{a}$$

• $F$: resultant force in N.
• $m$: mass in kg.
• $a$: acceleration in m/s², in the direction of the resultant force.

Relationships and symbols

• Constant mass: $a \propto F$. Double the resultant force → double the acceleration.
• Constant resultant force: $a \propto 1/m$. Double the mass → halve the acceleration.
• $\propto$ means ‘is proportional to’; $\sim$ means ‘approximately’.

Higher Tier: inertia and inertial mass

• Inertia: tendency to remain at rest or in uniform motion.
• Inertial mass: difficulty of changing velocity; defined by $m = F/a$.
• Same resultant force: smaller acceleration → larger inertial mass.