A body rotates about a fixed axis with an angular acceleration of one radian/second/second. Through what angle does it rotate during the time in which its angular velocity increases from 5 rad/s to 15 rad/s.

Explanation

To solve the problem, we will use the equations of motion for rotational dynamics. The key variables we have are:

- Initial angular velocity (\( \omega_0 \)) = 5 rad/s
- Final angular velocity (\( \omega \)) = 15 rad/s
- Angular acceleration (\( \alpha \)) = 1 rad/s²

We need to find the angular displacement (\( \theta \)) during the time the angular velocity increases from \( \omega_0 \) to \( \omega \).

### Step-by-Step Solution:

1. **Identify the known values**:
- Initial angular velocity, \( \omega_0 = 5 \, \text{rad/s} \)
- Final angular velocity, \( \omega = 15 \, \text{rad/s} \)
- Angular acceleration, \( \alpha = 1 \, \text{rad/s}^2 \)

2. **Use the rotational kinematic equation**:
We can use the equation:
\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]
Rearranging gives:
\[ \theta = \frac{\omega^2 - \omega_0^2}{2\alpha} \]

3. **Substitute the known values into the equation**:
\[ \theta = \frac{(15)^2 - (5)^2}{2 \times 1} \]

4. **Calculate the squares**:
\[ \theta = \frac{225 - 25}{2} \]

5. **Simplify the equation**:
\[ \theta = \frac{200}{2} = 100 \, \text{radians} \]

6. **Final answer**:
The angle through which the body rotates while its angular velocity increases from 5 rad/s to 15 rad/s is \( \theta = 100 \, \text{radians} \).