Santa loses his footing and slides down a frictionless, snowy roof that is tilted at an angle of 30.0. If Santa slides 5.00m before reaching the edge, what is his speed as he leaves the roof?
To determine Santa's speed as he leaves the roof, we can use the principles of conservation of energy and Newton's laws of motion.
First, let's break down the problem:
1. Santa is sliding down a frictionless, snowy roof.
2. The roof is tilted at an angle of 30.0 degrees.
3. Santa slides a distance of 5.00m before reaching the edge.
Since the roof is frictionless, the only force acting on Santa is gravity. Therefore, we can use the principle of conservation of energy, which states that the initial energy of an object is equal to its final energy.
The initial energy of Santa can be calculated as the potential energy at the top of the roof, and the final energy can be calculated as the kinetic energy just before he leaves the roof.
1. Initial energy (Ei) = Potential energy (PE) at the top of the roof
Ei = m * g * h, where m is Santa's mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the roof.
2. Final energy (Ef) = Kinetic energy (KE) just before leaving the roof
Ef = (1/2) * m * v^2, where v is the final velocity of Santa.
Since energy is conserved, Ei = Ef.
To find the final velocity (v), we need to determine the height of the roof (h). Since we know the roof is tilted at an angle of 30.0 degrees, we can use trigonometry to find the height:
h = 5.00m * sin(30.0)
Now we can substitute the values into the equation for initial energy (Ei):
Ei = m * g * h
Since Ei = Ef, we can equate the equations for initial and final energy:
m * g * h = (1/2) * m * v^2
Simplifying the equation:
g * h = (1/2) * v^2
To find v, we can solve for it:
v = sqrt(2 * g * h)
Now plug in the values into the equation and solve for v:
g = 9.8 m/s^2
h = 5.00m * sin(30.0)
Calculate h: h = 5.00m * 0.5 = 2.50m
v = sqrt(2 * 9.8 * 2.50)
v ≈ 7.91 m/s
Therefore, Santa's speed as he leaves the roof is approximately 7.91 m/s.
Use the vertical distance that he falls to compute his potential energy loss, which equals his kinetic energy gain.
Compute V from that