A skateboarder with an initial speed of 3.1-m/s, rolls virtually friction free down a straight incline of length 18-m in 3.2-s. At what angle (in degrees) is the incline oriented above the horizontal?
To find the angle of the incline, we can use the formula:
θ = arctan(h / L)
Where:
θ is the angle of the incline
h is the vertical height of the incline
L is the length of the incline
In this case, we have the length of the incline (L = 18 m). We need to find the vertical height (h).
Since the skateboarder rolls virtually friction-free, we can assume that there is no loss of energy due to friction. Therefore, the skateboarder's initial kinetic energy is converted into potential energy at the top of the incline.
The initial kinetic energy (KE) can be calculated using the formula:
KE = 0.5 * m * v^2
Where:
m is the mass of the skateboarder (which we assume to be negligible)
v is the initial speed of the skateboarder (v = 3.1 m/s)
So, the initial kinetic energy is 0.5 * v^2.
The potential energy (PE) at the top of the incline is given by the formula:
PE = m * g * h
Where:
m is the mass of the skateboarder
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the vertical height
Since we assume the mass of the skateboarder to be negligible, we can ignore the term m * g.
Therefore, we have:
0.5 * v^2 = h * g
Rearranging the equation, we get:
h = (0.5 * v^2) / g
Now, we can substitute the given values:
v = 3.1 m/s
g = 9.8 m/s^2
h = (0.5 * 3.1^2) / 9.8
Calculating further, we find:
h ≈ 0.494 m
Finally, we can substitute the values of h and L into the arctan formula to find the angle (θ):
θ = arctan(0.494 / 18)
Using a calculator, we find:
θ ≈ 1.56 degrees
Therefore, the incline is oriented approximately 1.56 degrees above the horizontal.
To determine the angle at which the incline is oriented above the horizontal, we can use the equation of motion for an object rolling down an incline without friction:
v = u + at
Where:
v = final velocity
u = initial velocity
a = acceleration
t = time
Since the skateboarder rolls virtually friction-free, the only force acting on them is due to gravity. Thus, the skateboarder experiences an acceleration due to gravity acting downhill.
Taking the positive direction of the incline as downward, the acceleration due to gravity can be calculated using the formula:
a = g * sin(θ)
Where:
g = acceleration due to gravity (approximately 9.8 m/s^2)
θ = angle of the incline
We can rearrange the first equation to solve for the acceleration:
a = (v - u) / t
Substituting the given values:
v = 0 m/s (since the skateboarder comes to a stop at the end of the incline)
u = 3.1 m/s
t = 3.2 s
a = (0 - 3.1) / 3.2
Now we can substitute this acceleration value into the equation for the acceleration due to gravity:
(0 - 3.1) / 3.2 = 9.8 * sin(θ)
-3.1 / 3.2 = 9.8 * sin(θ)
sin(θ) = -3.1 / (3.2 * 9.8)
Now, we can take the inverse sine (arcsin) of both sides to find the angle:
θ = arcsin(-3.1 / (3.2 * 9.8))
Using a calculator, we can determine the corresponding angle:
θ ≈ -15.4 degrees
Since the angle cannot be negative in this context, we take the positive value:
θ ≈ 15.4 degrees
Therefore, the incline is oriented at approximately 15.4 degrees above the horizontal.