Understanding the Physics

* Uniform Acceleration: A body sliding down a frictionless inclined plane experiences constant acceleration due to gravity. The acceleration component along the incline is *g*sin(θ), where *g* is the acceleration due to gravity (9.8 m/s²) and θ is the angle of inclination.

* Kinematics: We'll use the equations of motion to relate the distance traveled, acceleration, and time.

Steps

1. Define Variables:

* *s* = distance traveled (19.4 m)

* *t* = time (3 seconds) - Note that we're considering the *third* second, so we'll need to account for the distance traveled in the first two seconds.

* *a* = acceleration = *g*sin(θ)

* *θ* = angle of inclination (what we want to find)

2. Find the Distance Traveled in the First Two Seconds:

* Use the equation: *s* = *ut* + (1/2)*a*t²

* Initial velocity (*u*) is 0 since the body starts from rest.

* The acceleration (*a*) is *g*sin(θ).

* Time (*t*) is 2 seconds.

* Substitute and simplify: *s* = (1/2) * *g*sin(θ) * 2² = 2 * *g*sin(θ)

3. Find the Distance Traveled in the Third Second:

* The distance traveled in the third second is the total distance in three seconds minus the distance traveled in the first two seconds.

* *s* (third second) = 19.4 m - 2 * *g*sin(θ)

4. Apply the Equation of Motion for the Third Second:

* *s* (third second) = *u*t + (1/2)*a*t²

* *u* is the velocity at the beginning of the third second (which is the final velocity after the first two seconds).

* *t* is 1 second.

* *a* is *g*sin(θ)

5. Find the Velocity at the Beginning of the Third Second:

* *u* = *at* = *g*sin(θ) * 2 = 2 * *g*sin(θ)

6. Substitute and Solve for θ:

* 19.4 - 2 * *g*sin(θ) = (2 * *g*sin(θ)) * 1 + (1/2) * *g*sin(θ) * 1²

* 19.4 = (5/2) * *g*sin(θ)

* sin(θ) = (19.4 * 2) / (5 * 9.8)

* θ = arcsin(19.4 * 2 / (5 * 9.8))

* θ ≈ 22.6 degrees

Therefore, the angle of inclination of the plane is approximately 22.6 degrees.