(a)State and explain Newton's second law of motion.
(b) A 1000 kg vehicle moving with a speed of 20 m/s is brought to rest in a distance of 50 metres:
(i) Find the acceleration.
(ii) Calculate the unbalanced force acting on the vehicle.

Answer

### Step-by-Step Solution

#### (a) State and explain Newton's second law of motion.

**Newton's Second Law of Motion** states that the rate of change of momentum of an object is directly proportional to the net force acting on it. Mathematically, it can be expressed as:

\[ F = \frac{dp}{dt} \]

Where:
- \( F \) is the net force applied,
- \( p \) is the momentum of the object, defined as \( p = mv \) (mass times velocity),
- \( t \) is the time.

If we consider mass \( m \) to be constant, we can rewrite the equation as:

\[ F = m \cdot a \]

Where:
- \( a \) is the acceleration of the object.

This means that the force acting on an object is equal to the mass of the object multiplied by its acceleration. The direction of the force is the same as the direction of the acceleration.

#### (b) A 1000 kg vehicle moving with a speed of 20 m/s is brought to rest in a distance of 50 metres.

**Given Data:**
- Mass of the vehicle, \( m = 1000 \, \text{kg} \)
- Initial velocity, \( u = 20 \, \text{m/s} \)
- Final velocity, \( v = 0 \, \text{m/s} \) (since the vehicle is brought to rest)
- Distance, \( s = 50 \, \text{m} \)

**(i) Find the acceleration.**

We can use the kinematic equation:

\[ v^2 = u^2 + 2as \]

Substituting the known values:

\[ 0^2 = (20)^2 + 2a(50) \]

This simplifies to:

\[ 0 = 400 + 100a \]

Rearranging gives us:

\[ 100a = -400 \]

Thus,

\[ a = \frac{-400}{100} = -4 \, \text{m/s}^2 \]

**(ii) Calculate the unbalanced force acting on the vehicle.**

Using Newton's second law:

\[ F = m \cdot a \]

Substituting the values:

\[ F = 1000 \, \text{kg} \cdot (-4 \, \text{m/s}^2) \]

Calculating this gives:

\[ F = -4000 \, \text{N} \]

The negative sign indicates that the force is acting in the opposite direction to the motion of the vehicle.