A unicorn stands atop a 34° hill. The unicorn gracefully slides down the grassy hill with ì=.16. Calculate the acceleration of the unicorn as it begins to slide down the hill. (b) If the unicorn slides for 13.9s before reaching the bottom, how far did the unicorn slide?
To calculate the acceleration of the unicorn as it slides down the hill, we can use the components of the gravitational force acting on the unicorn.
First, let's break down the force acting on the unicorn into its components. The force of gravity can be divided into two components - one parallel to the slope of the hill and one perpendicular to it. The component parallel to the slope of the hill is responsible for the acceleration of the unicorn as it slides down.
The gravitational force acting on the unicorn is given by the formula:
Fg = mg
Where Fg is the force of gravity, m is the mass of the unicorn, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
To determine the component of the force parallel to the slope, we can use trigonometry. The force parallel to the slope is given by:
F_parallel = Fg * sin(θ)
Where θ is the angle of the slope (34° in this case).
Now, we can calculate the acceleration using Newton's second law, which relates force to mass and acceleration:
F_parallel = m * a
Rearranging the equation, we get:
a = F_parallel / m
Let's substitute the values into the equation:
F_parallel = Fg * sin(θ)
= mg * sin(θ)
a = (mg * sin(θ)) / m
= g * sin(θ)
Given that g is approximately 9.8 m/s^2 and θ is 34°, we can calculate the acceleration:
a = 9.8 * sin(34°)
Now, let's calculate the acceleration:
a ≈ 5.34 m/s^2
Therefore, the acceleration of the unicorn as it begins to slide down the hill is approximately 5.34 m/s^2.
Moving on to part (b) of the question, to calculate the distance the unicorn slides, we can use the equations of motion for constant acceleration.
The equation relating distance (d), initial velocity (v₀), time (t), and acceleration (a) is:
d = v₀ * t + (1/2) * a * t^2
Since the unicorn begins from rest (v₀ = 0), the equation simplifies to:
d = (1/2) * a * t^2
Given that the acceleration (a) is 5.34 m/s^2 and the time (t) is 13.9 s, we can substitute these values into the equation to calculate the distance:
d = (1/2) * 5.34 * (13.9)^2
Now, let's calculate the distance:
d ≈ 517.43 meters
Therefore, the unicorn slides approximately 517.43 meters before reaching the bottom of the hill.