A crate is given an initial speed of 5.0m/s up the 22.5∘ plane shown in (Figure 1) . Assume μk = 0.12.
To answer this question, we need to find the acceleration of the crate on the inclined plane and then use that to calculate the distance it will travel.
First, let's calculate the acceleration of the crate on the inclined plane. The formula for the acceleration of an object on an inclined plane is:
a = g * sin(θ) - μk * g * cos(θ)
where:
a = acceleration
g = acceleration due to gravity (approximately 9.8 m/s^2)
θ = angle of the inclined plane (22.5° in this case)
μk = coefficient of kinetic friction (0.12 in this case)
Substituting the given values into the formula, we have:
a = (9.8 m/s^2) * sin(22.5°) - (0.12) * (9.8 m/s^2) * cos(22.5°)
Now, let's calculate the acceleration:
a = 3.385 m/s^2
Next, we can use the equation of motion to find the distance traveled by the crate. The equation we'll use is:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s, as the crate comes to a stop)
u = initial velocity (5.0 m/s)
a = acceleration (3.385 m/s^2)
s = distance traveled (what we're solving for)
Substituting the given values into the equation, we have:
0^2 = (5.0 m/s)^2 + 2 * (3.385 m/s^2) * s
Simplifying the equation, we get:
0 = 25.0 m^2/s^2 + 6.77 m/s^2 * s
Rearranging the equation, we have:
-25.0 m^2/s^2 = 6.77 m/s^2 * s
Dividing both sides of the equation by 6.77 m/s^2, we get:
s = -25.0 m^2/s^2 / 6.77 m/s^2
s ≈ -3.69 m
Since distance cannot be negative, we can ignore the negative sign. Therefore, the distance traveled by the crate is approximately 3.69 meters up the inclined plane.