**Here is a lengthy example of a radical equation used in physics:**

*One common radical equation used in physics involves calculating displacement given initial position, initial velocity, acceleration, and time. To solve for displacement (d) given these variables, we can set up and solve the following radical equation:*

d = x0 + v0t + (1/2)at^2

Where:

x0 is the initial position

v0 is the initial velocity

a is the acceleration

t is the time

**Let’s think through what each term in this equation represents:**

- x0 represents the object’s starting position. This gives us the baseline or initial location of the object.
- v0t represents the displacement due only to the object’s initial velocity over time. We multiply the initial velocity v0 by the time t to determine how far the object would travel if acceleration was 0.
- (1/2)at^2 represents the additional displacement caused by acceleration. Acceleration causes the velocity to change over time. We can use the physics formula that describes how velocity changes with constant acceleration (v = v0 + at) and integrate it to determine displacement. The integral of this formula is d = x0 + v0t + (1/2)at^2.
- By adding all three terms together – the initial position x0, the displacement due to initial velocity v0t, and the additional displacement due to acceleration (1/2)at^2 – we can calculate the total displacement d of an object experiencing constant acceleration starting from an initial position and velocity after some time t.

**To see a full example of using this radical equation, let’s suppose we have:**

- Initial position x0 = 5 meters
- Initial velocity v0 = 2 m/s
- Acceleration a = 3 m/s^2
- Time t = 4 seconds

**To solve for the total displacement d, we plug these values into the equation:**

d = x0 + v0t + (1/2)at^2

d = 5 m + (2 m/s)(4 s) + (1/2)(3 m/s^2)(4 s)^2

d = 5 m + 8 m + 24 m

d = 37 meters

So in this example, an object starting 5 meters from the origin, with an initial velocity of 2 m/s, experiencing an acceleration of 3 m/s^2 for 4 seconds would have a total displacement of 37 meters.

*This radical equation involving square roots is commonly used in physics problems involving kinematics, or the motion of objects, to analytically solve for variables like displacement, velocity, or acceleration given appropriate starting information and relationships. It provides a clear and systematic way to break down an object’s motion due to different causes and combine them to determine the total effect. Does this help explain an example of how such a radical equation is applied and used in physics?? Let me know if you need any clarification or have additional questions!*