Newton's second law

Force, mass and acceleration

Newton's second law of motion can be described by this equation:

resultant force = mass × acceleration

\text{F} = \text{ma}

This is when:

  • Force ( \text{F}) is measured in newtons (N)
  • Mass ( \text{m}) is measured in kilograms (kg)
  • Acceleration ( \text{a}) is measured in metres per second squared (m/s²)

The equation shows that the acceleration of an object is:

In other words, the acceleration of an object increases if the resultant force on it increases, and decreases if the mass of the object increases.

Example

Calculate the force needed to accelerate a 22 kg cheetah at 15 m/s².

\text{F} = \text{ma}

= 22 × 15

= 330 N

Question

Calculate the force needed to accelerate a 15 kg gazelle at 10 m/s².

\text{F} = \text{ma}

= 15 × 10

= 150 N

Estimations

It is important to be able to estimate speeds, accelerations and forces involved in road vehicles. The symbol ~ is used to indicate that a value or answer is an approximate one. The table gives some examples.

VehicleMaximum legal speed on a single carriageway in m/sMass in kgAcceleration in m/s2
Family car~27~1,600~3
Lorry~22~36,000~0.4

Example

Estimate the force needed to accelerate a family car to its top speed on a single carriageway.

Using values of ~1,600 kg and ~3 m/s2, and \text{F} = \text{ma}:

1,600 × 3 = ~4,800 N

Question

Estimate the force needed to accelerate a lorry to its top speed on a single carriageway.

Using values of ~36,000 kg and ~0.4 m/s2, and \text{F} = \text{ma}:

Force ( \text{F}) is ~14,400 N

Acceleration when considering more than one force

Acceleration occurs if there is a resultant force. If there is only one force, then that is the resultant force. But if there are two or more forces acting, it is important to realise that of all the acting forces are resultant, which causes a body to accelerate.

Example

A 2 kg box of biscuits is pushed across a table with a force of 10 N. There is a frictional force of 4 N. What will be the acceleration?

First find the resultant force: This will be 6 N because the 10 N and 4 N forces oppose each other.

Next use \text{F} = \text{ma} but rearrange this to find acceleration.

\text{F} = \text{ma} becomes \text{a} = \frac{\text{F}}{\text{m}}

Acceleration = 6 ÷ 2 = 3 m/s2

Question

A 200 kg motorbike experiences a thrust of 100 N and total resistive forces of 50 N. What would be the acceleration?

Resultant force = 100 – 50 = 50 N

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

acceleration = 50 ÷ 200 = 0.25 m/s2