# 5.7.2 Newton's Second Law

### 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: 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: 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.

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 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. 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 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.

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: 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

• 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

#### Exam Tip

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. ### Author: Jonathan

Jonathan graduated with a first-class Master's degree in Theoretical Physics from Imperial College London. He has worked in education for more than a decade as a Maths and Physics Teacher, Tutor, Head of Physics, and most recently, as Assistant Headteacher. He is now an Educational Consultant and works with us to design and improve our Physics resources.
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