A car starts accelerating with an acceleration of `(1)/(3)m//s^(2)` when its speed was `18 km//h`. If the desplacement of car in `n^(th)` minute after it starts accelerating is `4.8km` then n is :

Explanation

To solve the problem step by step, we will follow the given information and apply the relevant equations of motion.

### Step 1: Convert Initial Speed to m/s
The initial speed \( u \) is given as \( 18 \, \text{km/h} \). We need to convert this to meters per second (m/s).

\[ u = 18 \, \text{km/h} = \frac{18 \times 1000 \, \text{m}}{3600 \, \text{s}} = 5 \, \text{m/s} \]

**Hint:** Remember that to convert km/h to m/s, multiply by \( \frac{1000}{3600} \).

### Step 2: Convert Acceleration to m/min²
The acceleration \( a \) is given as \( \frac{1}{3} \, \text{m/s}^2 \). We want to convert this to meters per minute squared (m/min²).

\[ a = \frac{1}{3} \, \text{m/s}^2 = \frac{1}{3} \times 60^2 \, \text{m/min}^2 = \frac{1}{3} \times 3600 \, \text{m/min}^2 = 1200 \, \text{m/min}^2 \]

**Hint:** To convert m/s² to m/min², multiply by \( 60^2 \).

### Step 3: Use the Displacement Formula
We know that the displacement \( s \) in the \( n \)th minute can be calculated using the formula:

\[ s = u \cdot n + \frac{1}{2} a (n^2 - (n-1)^2) \]

The term \( (n^2 - (n-1)^2) \) simplifies to \( 2n - 1 \).

Substituting the values we have:

\[ s = 5n + \frac{1}{2} \cdot 1200 \cdot (2n - 1) \]

### Step 4: Set Up the Equation
We know the displacement \( s \) is \( 4.8 \, \text{km} = 4800 \, \text{m} \).

Setting the equation:

\[ 4800 = 5n + \frac{1}{2} \cdot 1200 \cdot (2n - 1) \]

### Step 5: Simplify the Equation
Now simplify the equation:

\[ 4800 = 5n + 600(2n - 1) \]
\[ 4800 = 5n + 1200n - 600 \]
\[ 4800 + 600 = 1205n \]
\[ 5400 = 1205n \]

### Step 6: Solve for \( n \)
Now, solve for \( n \):

\[ n = \frac{5400}{1205} \approx 4.48 \]

Since \( n \) must be a whole number, we round down to \( n = 4 \).

### Final Answer
Thus, the value of \( n \) is:

\[ \boxed{4} \]

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