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 step by step, we will analyze the motion of the electric car in three phases: acceleration, constant speed, and deceleration.

### Step 1: Calculate the time to reach maximum speed
The car starts from rest, so the initial velocity \( u = 0 \). The maximum speed \( V = 20 \, \text{m/s} \) and the acceleration \( a = 1 \, \text{m/s}^2 \).

Using the first equation of motion:
\[ V = u + at \]
Substituting the values:
\[ 20 = 0 + 1 \cdot t_1 \]
Thus:
\[ t_1 = 20 \, \text{s} \]

### Step 2: Calculate the distance covered during acceleration
Using the second equation of motion:
\[ s_1 = ut + \frac{1}{2} a t^2 \]
Substituting the values:
\[ s_1 = 0 \cdot 20 + \frac{1}{2} \cdot 1 \cdot (20)^2 \]
Calculating:
\[ s_1 = \frac{1}{2} \cdot 1 \cdot 400 = 200 \, \text{m} \]

### Step 3: Calculate the time to decelerate to rest
The initial velocity for deceleration is \( u = 20 \, \text{m/s} \) and the deceleration \( a = 2 \, \text{m/s}^2 \). The final velocity \( V = 0 \).

Using the first equation of motion:
\[ V = u - at \]
Substituting the values:
\[ 0 = 20 - 2t_2 \]
Thus:
\[ t_2 = 10 \, \text{s} \]

### Step 4: Calculate the distance covered during deceleration
Using the second equation of motion:
\[ s_2 = ut - \frac{1}{2} a t^2 \]
Substituting the values:
\[ s_2 = 20 \cdot 10 - \frac{1}{2} \cdot 2 \cdot (10)^2 \]
Calculating:
\[ s_2 = 200 - 100 = 100 \, \text{m} \]

### Step 5: Calculate the remaining distance to cover
The total distance is \( 1 \, \text{km} = 1000 \, \text{m} \).

The distance covered during acceleration and deceleration:
\[ s_1 + s_2 = 200 + 100 = 300 \, \text{m} \]

Remaining distance \( s_3 \):
\[ s_3 = 1000 - 300 = 700 \, \text{m} \]

### Step 6: Calculate the time to cover the remaining distance at maximum speed
At maximum speed \( V = 20 \, \text{m/s} \):
\[ t_3 = \frac{s_3}{V} = \frac{700}{20} = 35 \, \text{s} \]

### Step 7: Calculate the total time
The total time \( T \) is:
\[ T = t_1 + t_2 + t_3 = 20 + 10 + 35 = 65 \, \text{s} \]

### Step 8: Relate total time to \( 13n \)
According to the problem, the total time is given as \( 13n \):
\[ 65 = 13n \]
Thus:
\[ n = \frac{65}{13} = 5 \]

### Final Answer
The value of \( n \) is \( 5 \).

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