Children sled down a 45 m long hill inclined at 26 degrees. At the bottom, the slope levels out. If the coefficient of friction is 0.1, how far do the children slide on the level ground?
s₁=45 m
α=26⁰
s₂=?
ΔPE +W(fr)₁ =KE =W(fr)₂,
ΔPE +W(fr)₁ =W(fr)₂,
mgh+ F(fr)₁•s₁ = F(fr)₂•s₂.
h=s₁•sinα,
F(fr)₁ =μ•N₁=μ•mgcosα,
F(fr)₂ =μ•N₂=μ•mg,
mgs₁sinα + μ•mgcosα•s ₁ = μ•mg s₂,
s₁•sinα + μ•cosα•s ₁ = μ•s₂,
s₂=(s₁•sinα + μ•cosα•s ₁)/μ =…
To calculate how far the children slide on the level ground, we need to consider the forces acting on them as they move down the hill.
Initially, the only force acting on the children is the force of gravity pulling them downward along the incline. This force can be resolved into two components: one parallel to the incline (msinθ) and another perpendicular to the incline (mcosθ), where m is the mass of the children and θ is the angle of the incline.
The force parallel to the incline is what causes the children to accelerate down the hill. This force can be calculated using the formula:
F_parallel = m * g * sinθ,
where g is the acceleration due to gravity (approximately 9.8 m/s²).
The force perpendicular to the incline is what creates the friction that slows the children down. This force can be calculated using the formula:
F_friction = m * g * cosθ * μ,
where μ is the coefficient of friction (given as 0.1 in this case).
Now, we know that when the children reach the bottom of the hill, the slope levels out, which means the incline is no longer present. Therefore, the force parallel to the incline becomes zero, and the only force acting on the children is the force of friction.
To find the distance the children slide on the level ground, we need to calculate the work done by the force of friction. The work done is equal to the force of friction multiplied by the distance.
The equation for work is given by:
Work = force * distance.
Rearranging the equation, we have:
Distance = Work / force.
Now, the work done by the force of friction can be calculated using the formula:
Work = F_friction * distance_on_level_ground.
Finally, we substitute the value of F_friction and solve for distance_on_level_ground:
distance_on_level_ground = (m * g * cosθ * μ * distance_on_level_ground) / (m * g * cosθ).
By cancelling out the common factors (m, g, cosθ), we get:
distance_on_level_ground = μ * distance_on_level_ground.
Since the distance on the level ground is the same as the length of the hill (45 m), we can substitute the value and solve for distance_on_level_ground:
distance_on_level_ground = 0.1 * 45.
Therefore, the children slide for a distance of 4.5 meters on the level ground.