A small electric car has a maximum constant acceleration of `1m//s^(2)`, a maximum constant deceleration of `2m//s^(2)` and a maximum speed of `20m//s`. The amount of minimum time it would take to drive this car `1km` starting from rest is `(13n)` second. Find value of `n`

Answer

To solve the problem, we need to determine the minimum time it would take for the electric car to travel a distance of 1 km (1000 m) starting from rest, given its maximum acceleration, deceleration, and maximum speed.

### Step-by-Step Solution:

1. **Identify the Parameters:**
- Maximum acceleration, \( a = 1 \, \text{m/s}^2 \)
- Maximum deceleration, \( d = 2 \, \text{m/s}^2 \)
- Maximum speed, \( v_{\text{max}} = 20 \, \text{m/s} \)
- Distance to travel, \( s = 1000 \, \text{m} \)

2. **Calculate the Time to Reach Maximum Speed:**
To find the time taken to reach the maximum speed, we use the formula:
\[ v = u + at \]
where \( u = 0 \) (starting from rest), \( v = v_{\text{max}} \), and \( a = 1 \, \text{m/s}^2 \).
\[ 20 = 0 + 1 \cdot t \implies t = 20 \, \text{s} \]

3. **Calculate the Distance Covered While Accelerating:**
The distance covered during acceleration can be calculated using:
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting \( u = 0 \), \( a = 1 \, \text{m/s}^2 \), and \( t = 20 \, \text{s} \):
\[ s = 0 + \frac{1}{2} \cdot 1 \cdot (20)^2 = \frac{1}{2} \cdot 1 \cdot 400 = 200 \, \text{m} \]

4. **Calculate Remaining Distance After Acceleration:**
The total distance to travel is 1000 m, and the distance covered during acceleration is 200 m. Therefore, the remaining distance is:
\[ s_{\text{remaining}} = 1000 - 200 = 800 \, \text{m} \]

5. **Calculate Time to Cover Remaining Distance at Maximum Speed:**
Since the car reaches its maximum speed of 20 m/s, the time to cover the remaining distance of 800 m is:
\[ t = \frac{s}{v} = \frac{800}{20} = 40 \, \text{s} \]

6. **Calculate Total Time Taken:**
The total time taken to travel 1 km is the sum of the time taken to accelerate and the time taken to travel the remaining distance:
\[ t_{\text{total}} = t_{\text{acceleration}} + t_{\text{constant speed}} = 20 + 40 = 60 \, \text{s} \]

7. **Convert Total Time to Required Form:**
The problem states that the minimum time taken is \( 13n \) seconds. Therefore, we set up the equation:
\[ 60 = 13n \implies n = \frac{60}{13} \approx 4.615 \]
Since \( n \) must be an integer, we round down to the nearest integer, which gives \( n = 4 \).

### Final Answer:
The value of \( n \) is \( 4 \).