Santa loses his footing and slides down a frictionless, snowy roof that is tilted at an angle of 30.0. If Santa slides 5.00 before reaching the edge, what is his speed as he leaves the roof?

We can use the conservation of energy principle to solve this problem. We'll equate the initial potential energy (when Santa is at the top of the roof) with the final kinetic energy (when Santa is at the edge of the roof).

Initially, Santa is at a height h above the ground. We can find this height using trigonometry with the given roof angle and the distance he slides.

h = 5.00 m * sin(30.0°)

h ≈ 2.50 m

The potential energy (PE) at the top is given by:

PE_initial = m * g * h

And the final kinetic energy (KE) is given by:

KE_final = 1/2 * m * v^2

Equating initial potential energy with final kinetic energy:

m * g * h = 1/2 * m * v^2

Where m is the mass of Santa, g is the acceleration due to gravity (9.81 m/s²), and v is the final speed we want to find. The mass of Santa will cancel out:

g * h = 1/2 * v^2

Now we can solve for v:

v^2 = 2 * g * h
v^2 ≈ 2 * 9.81 m/s² * 2.50 m
v^2 ≈ 49.05 m²/s²

Taking the square root of both sides:

v ≈ 7.00 m/s

So Santa's speed as he leaves the roof is approximately 7.00 m/s.

To find Santa's speed as he leaves the roof, we can use the principles of kinematics. Here are the steps to calculate his speed:

Step 1: Determine the acceleration
Since the roof is frictionless, the only force acting on Santa is the component of his weight parallel to the slope. This force can be calculated as F = m * g * sin(θ), where m is Santa's mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and θ is the angle of the roof tilt. So, the acceleration is a = g * sin(θ).

Step 2: Calculate the time taken
Using the kinematic equation vf = vi + a * t, where vf is the final velocity, vi is the initial velocity, a is the acceleration, and t is the time taken, we can rearrange the equation to solve for t: t = (vf - vi) / a.

The initial velocity (vi) is 0 since Santa starts from rest. The final velocity (vf) is what we need to find.

Step 3: Substitute the known values
Substituting the known values into the equation, we have t = (vf - 0) / a, which simplifies to t = vf / a. We also know that Santa slides a distance of 5.00 m before reaching the edge.

Step 4: Calculate the speed
To find the time taken (t), we can use the kinematic equation x = vi * t + (1/2) * a * t^2, where x is the distance traveled, vi is the initial velocity, a is the acceleration, and t is the time taken.

Substituting the known values, we have 5.00 = 0 * t + (1/2) * a * t^2. Since the initial velocity is 0, the equation simplifies to 5.00 = (1/2) * a * t^2.

Step 5: Rearrange the equation
Rearranging the equation, we have t^2 = (2 * 5.00) / a.

Step 6: Calculate the time taken
Taking the square root of both sides of the equation, we get t = √[(2 * 5.00) / a].

Step 7: Calculate the final velocity
Finally, substituting the values of t and a into the equation t = vf / a, we get √[(2 * 5.00) / a] = vf / a. Rearranging the equation, we have vf = √(2 * 5.00 * a).

Substituting the value of a = g * sin(θ) obtained in Step 1, we can calculate the final velocity:

vf = √(2 * 5.00 * g * sin(θ))

Substituting the value of the angle θ = 30.0 degrees, and the acceleration due to gravity g = 9.8 m/s^2, the calculation would be:

vf = √(2 * 5.00 * 9.8 * sin(30.0))

Evaluating the above expression, we find:

vf ≈ 8.05 m/s

Therefore, Santa's speed as he leaves the roof is approximately 8.05 m/s.

To find Santa's speed as he leaves the roof, we can use the principles of physics, specifically the component of gravity acting parallel to the roof's surface. Let's break it down step by step:

1. Determine the gravitational force parallel to the roof: The gravitational force acting on Santa can be divided into two components: one perpendicular to the roof and one parallel to the roof. Since there is no friction on the roof, the only force acting parallel to the roof is the component of gravity. We can calculate this component using the equation:
F_parallel = m * g * sin(θ),
where m is Santa's mass, g is the acceleration due to gravity (9.8 m/s^2), and θ is the angle of the roof (30.0 degrees).

2. Calculate the work done by gravity: As Santa slides down the roof, gravity does work on him, which is given by the equation:
Work = F_parallel * d,
where F_parallel is the parallel component of gravity we calculated in the previous step, and d is the distance Santa slides (5.00 m).

3. Equate the work done by gravity to the change in Santa's kinetic energy: According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy. In this case, we can express this as:
Work = ΔKE,
where ΔKE is the change in kinetic energy.

4. Calculate the kinetic energy: The kinetic energy of an object is given by the equation:
KE = (1/2) * m * v^2,
where m is the mass of Santa and v is his speed.

5. Solve for v: By equating the work done by gravity to the change in kinetic energy, we get:
F_parallel * d = ΔKE.
Substituting the equations from steps 1 and 4, we have:
m * g * sin(θ) * d = (1/2) * m * v^2.
Simplifying and solving for v, we find:
v = sqrt(2 * g * sin(θ) * d).

Now, you can plug in the values and calculate Santa's speed as he leaves the roof.