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Newton’s Second Law (AQA GCSE Combined Science: Synergy: Physical Sciences): Revision Note

Exam code: 8465

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Newton's Second Law

  • Newton's second law of motion states:

    The acceleration of an object is proportional to the resultant force acting on it and inversely proportional to the object's mass

  • Newton's second law explains the following important principles:

    • An object will accelerate (change its velocity) in response to a resultant force

    • The bigger this resultant force, the larger the acceleration

    • For a given force, the greater the object's mass, the smaller the acceleration experienced

  • The image below shows some examples of Newton's second law in action:

Diagram showing two examples of Newton's Second Law: a baseball being struck by a bat accelerates due to the resultant force, and a lawnmower being pushed accelerates in the direction of the applied force
Objects like baseballs and lawnmowers accelerate when a resultant force is applied on them. The size of the acceleration is proportional to the size of the resultant force

Calculating Force & Acceleration

  • Newton's second law can be expressed as an equation:

F = ma

  • Where:

    • F = resultant force on the object in Newtons (N)

    • m = mass of the object in kilograms (kg)

    • a = acceleration of the object in metres per second squared (m/s2)

  • This equation can be rearranged with the help of a formula triangle:

Formula triangle for force, mass and acceleration: force (F) at the top, mass (m) and acceleration (a) at the bottom
Force, mass, acceleration formula triangle

Worked Example

A car salesman says that his best car has a mass of 900 kg and can accelerate from 0 to 27 m/s in 3 seconds.

Calculate:

a) The acceleration of the car in the first 3 seconds.

b) The force required to produce this acceleration.

Answer:

Part (a)

Step 1: List the known quantities

  • Initial velocity = 0 m/s

  • Final velocity = 27 m/s

  • Time, t = 3 s

Step 2: Calculate the change in velocity

change in velocity = Δv = final velocity − initial velocity

Δv = 27 − 0 = 27 m/s

Step 3: State the equation for acceleration

a = fraction numerator capital delta v over denominator t end fraction

Step 4: Calculate the acceleration

a = 27 ÷ 3 = 9 m/s2

Part (b)

Step 1: List the known quantities

  • Mass of the car, m = 900 kg

  • Acceleration, a = 9 m/s2

Step 2: Identify which law of motion to apply

  • The question involves quantities of force, mass and acceleration, so Newton's second law is required:

F = ma

Step 3: Calculate the force required to accelerate the car

F = 900 × 9 = 8100 N

Worked Example

Three shopping trolleys, A, B and C, are being pushed using the same force. This force causes each trolley to accelerate.

Diagram of three shopping trolleys A, B and C being pushed with the same force: trolley A is lightest, B is heaviest, C is intermediate mass

Which trolley will have the smallest acceleration? Explain your answer.

Answer:

Step 1: Identify which law of motion to apply

  • The question involves quantities of force and acceleration, and the image shows trolleys of different masses, so Newton's second law is required:

F = ma

Step 2: Re-arrange the equation to make acceleration the subject

a = F over m

Step 3: Explain the inverse proportionality between acceleration and mass  

  • Acceleration is inversely proportional to mass

  • This means for the same amount of force, a large mass will experience a small acceleration

  • Therefore, trolley C will have the smallest acceleration because it has the largest mass

Estimating Speed, Acceleration & Force

  • Newton's second law can be used to estimate the sizes of forces and accelerations in realistic scenarios

  • When estimating quantities, an approximate answer can be shown using the symbol ~

    • For example, an adult person has a mass of ~70 kg

Worked Example

A passenger travels in a car at a moderate speed. The vehicle is involved in a collision, which brings the car (and the passenger) to a halt in 0.1 seconds.

Estimate:

a) The acceleration of the car (and the passenger).

b) The force on the passenger.

Answer:

Part (a)

Step 1: Estimate the required quantities and list the known quantities

A moderate speed for a car is about 50 mph or 20 m/s

  • Initial velocity ~ 20 m/s

  • Final velocity = 0 m/s

  • Time, t = 0.1 s

Step 2: Calculate the change in velocity of the car (and the passenger)

change in velocity = Δv = final velocity − initial velocity

Δv = 0 − 20

Δv = −20 m/s

Step 3: Calculate the acceleration of the car (and the passenger) using the equation:

a = fraction numerator capital delta v over denominator t end fraction

Step 4: Calculate the deceleration

a = −20 ÷ 0.1

a = ~ −200 m/s2

Part (b)

Step 1: Estimate the required quantities and list the known quantities

An adult person has a mass of about 70 kg

  • Mass of the passenger, m ~ 70 kg

  • Acceleration, a = −200 m/s2

Step 2: State Newton's second law 

  • This question involves quantities of force, mass and acceleration, so the equation for Newton's second law is:

F = ma

Step 3: Calculate an estimate for the decelerating force

F = 70 × −200

F ~ −14 000 N

Examiner Tips and Tricks

Remember that resultant force is a vector quantity.

Examiners may ask you to comment on why its value is negative - this happens when the resultant force acts in the opposite direction to the object's motion.

In the worked example above, the resultant force opposes the passenger's motion, slowing them down (decelerating them) to a halt, this is why it has a minus symbol.

Inertia

Higher Tier Only

  • The concept of inertia is closely related to motion, it is defined as:

    The tendency of an object to continue in its state of rest, or in uniform motion unless acted upon by an external force

  • In other words, inertia is an object's resistance to a change in motion 

    • If an object is at rest, it will tend to remain at rest

    • If an object is moving at a constant velocity (constant speed in a straight line), it will continue to do so

  • The image below illustrates the concept of inertia using a coin and a cup of water:

Diagram demonstrating inertia using a coin on a card placed over a cup: when the card is flicked away quickly, the coin drops into the cup because its inertia keeps it stationary
Demonstrating the inertia of a small coin

Inertial Mass

  • Inertial mass is the property of an object which describes how difficult it is to change its velocity

  • It is defined as the ratio between the force applied to it and the acceleration it experiences, and can be calculated by:

i n e r t i a l space m a s s space equals space fraction numerator f o r c e over denominator a c c e l e r a t i o n end fraction

m space equals space F over a

  • Where:

    • m = inertial mass in kilograms (kg)

    • F = force in newtons (N)

    • a = acceleration in metres per second squared (m/s2)

  • This equation shows that for a given force, inertial mass is inversely proportional to acceleration

  • This means that:

    • Larger inertial masses are harder to accelerate, so they speed up more slowly

    • Smaller inertial masses are easier to accelerate, so they speed up more quickly

Diagram showing a small object being pushed with a force, resulting in a large acceleration, to illustrate that smaller inertial mass leads to greater acceleration for the same force
Diagram showing a large object being pushed with the same force, resulting in a smaller acceleration, to illustrate that larger inertial mass leads to less acceleration for the same force

This collection of objects have a variety of inertial masses. For the same applied force, their accelerations are inversely proportional to their mass

Worked Example

Three objects are used by a physics technician to demonstrate the concept of inertial mass. She applies the same force to each of the three objects and notes that:

  • Object A accelerates at 1.5 m/s2

  • Object B accelerates at 0.7 m/s2

  • Object C accelerates at 2.0 m/s2

Which object has the largest inertial mass?

Answer:

Step 1: State the definition of inertial mass

  • Inertial mass is defined as the ratio of force to acceleration:

inertial mass = force over acceleration

Step 2: Apply the definition to each of the objects in the question

  • Since the same force is applied to all three objects, the inertial mass of each object depends on how much it accelerates

    • An object that accelerates less must have a greater inertial mass.

  • This means the object with the smallest acceleration will have the largest inertial mass.

  • Object B has the smallest acceleration (0.7 m/s²), so it has the largest inertial mass.

Examiner Tips and Tricks

The definition of inertial mass as a ratio between force and acceleration should look similar to Newton's second law. That's because when you calculate the mass of an object using Newton's second law, you are in fact calculating its inertial mass.

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Ashika

Author: Ashika

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