A brick of mass 0.6 kg slides down an icy roof inclined at 30.0° with respect to the horizontal.

(a) If the brick starts from rest, how fast is it moving when it reaches the edge of the roof 1.75 m away? Ignore friction.
m/s

(b) Redo part (a) if the coefficient of kinetic friction is 0.18.

To solve both parts (a) and (b), we can use the principles of kinematics and the laws of motion. We'll break down the problem into smaller steps:

Step 1: Calculate the gravitational force acting on the brick.
The gravitational force (also known as weight) acting on an object is given by the formula:
Weight = mass × gravity
where mass is the mass of the object (given as 0.6 kg) and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).

Step 2: Resolve the gravitational force along the incline.
Since the roof is inclined at 30.0° with respect to the horizontal, we need to resolve the gravitational force into components parallel and perpendicular to the incline. The component parallel to the incline is given by the formula:
Component_parallel = Weight × sin(θ)
where θ is the angle of incline (30.0°).

Step 3: Calculate the acceleration along the incline.
When there is no friction, the acceleration along the incline can be determined using the formula:
Acceleration = Component_parallel / mass

Step 4: Calculate the final velocity of the brick.
To find the final velocity of the brick when it reaches the edge of the roof, we can use the kinematic equation:
Final velocity^2 = Initial velocity^2 + 2 × acceleration × distance
Since the brick starts from rest (initial velocity is 0 m/s), we can simplify the equation to:
Final velocity = √(2 × acceleration × distance)

(a) Let's now calculate the final velocity of the brick when there is no friction.

1. Calculate the gravitational force:
Weight = mass × gravity
= 0.6 kg × 9.8 m/s^2

2. Resolve the gravitational force along the incline:
Component_parallel = Weight × sin(θ)

3. Calculate the acceleration along the incline:
Acceleration = Component_parallel / mass

4. Calculate the final velocity:
Final velocity = √(2 × acceleration × distance)
= √(2 × Acceleration × 1.75 m)

Plug in the calculated values to find the final velocity in m/s.

(b) For part (b), where there is friction, we need to consider the additional force acting on the brick due to kinetic friction.
The force of kinetic friction can be calculated using the formula:
Force_friction = coefficient_friction × normal_force
where the normal force is equal to the weight of the brick (since it is on an incline and not on a vertical surface).

1. Calculate the gravitational force:
Weight = mass × gravity
= 0.6 kg × 9.8 m/s^2

2. Calculate the force of kinetic friction:
Force_friction = coefficient_friction × normal_force
= coefficient_friction × Weight

3. Resolve the gravitational force along the incline:
Component_parallel = Weight × sin(θ)

4. Calculate the net force along the incline:
Net force = Component_parallel - Force_friction

5. Calculate the acceleration along the incline:
Acceleration = Net force / mass

6. Calculate the final velocity:
Final velocity = √(2 × acceleration × distance)
= √(2 × Acceleration × 1.75 m)

Plug in the calculated values to find the final velocity in m/s.