A car is moving on circular path of radius 100 m such that its speed is increasing at the rate of `5(m)/(s^(2))` at `t=0` it starts from rest. The radial acceleration of car at the instant it makes one compelete round trip, will be equal to `10pix(m)/(s^(2))` then find x.

Answer

To solve the problem step by step, we will follow the information provided in the question and the video transcript.

### Step 1: Identify Given Information
- Radius of the circular path, \( R = 100 \, \text{m} \)
- Tangential acceleration, \( a_t = 5 \, \text{m/s}^2 \)
- Initial speed, \( u = 0 \, \text{m/s} \) (starts from rest)

### Step 2: Calculate Angular Acceleration
The angular acceleration \( \alpha \) can be calculated using the formula:
\[ \alpha = \frac{a_t}{R} \]
Substituting the values:
\[ \alpha = \frac{5 \, \text{m/s}^2}{100 \, \text{m}} = \frac{1}{20} \, \text{rad/s}^2 \]

### Step 3: Determine the Total Angle Rotated
Since the car makes one complete round trip, the angle rotated \( \theta \) in radians is:
\[ \theta = 2\pi \, \text{radians} \]

### Step 4: Use the Angular Motion Equation
Using the equation of motion for angular displacement:
\[ \omega^2 = u^2 + 2\alpha\theta \]
Since \( u = 0 \):
\[ \omega^2 = 0 + 2 \cdot \frac{1}{20} \cdot 2\pi \]
Calculating this gives:
\[ \omega^2 = \frac{2 \cdot 2\pi}{20} = \frac{4\pi}{20} = \frac{\pi}{5} \]
Thus,
\[ \omega = \sqrt{\frac{\pi}{5}} \, \text{rad/s} \]

### Step 5: Calculate Radial Acceleration
The radial (centripetal) acceleration \( a_r \) is given by the formula:
\[ a_r = R\omega^2 \]
Substituting the values:
\[ a_r = 100 \cdot \left(\sqrt{\frac{\pi}{5}}\right)^2 = 100 \cdot \frac{\pi}{5} = 20\pi \, \text{m/s}^2 \]

### Step 6: Equate to Find \( x \)
According to the problem, the radial acceleration is also given as:
\[ a_r = 10\pi x \]
Setting the two expressions for radial acceleration equal to each other:
\[ 20\pi = 10\pi x \]
Dividing both sides by \( 10\pi \):
\[ 2 = x \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{2} \]