A 3.6 kg wood block is launched up a wooden ramp that is inclined at a 30¡ angle. The block's initial speed is 13 m/s. (Use µk = 0.20 for the coefficient of kinetic friction for wood on wood.)

What vertical height does the block reach above its starting point?

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To find the vertical height reached by the block above its starting point, we need to analyze the forces acting on the block and calculate the work done.

1. Calculate the normal force acting on the block:
- The normal force is equal to the weight of the block acting perpendicular to the ramp.
- The weight of the block can be calculated as: weight = mass * gravity, where mass = 3.6 kg and gravity = 9.8 m/s^2.
- Weight = 3.6 kg * 9.8 m/s^2 = 35.28 N.
- The normal force is equal in magnitude but opposite in direction to the weight, so the normal force is also 35.28 N.

2. Calculate the force of friction acting on the block:
- The force of friction can be calculated as: force of friction = coefficient of kinetic friction * normal force.
- The coefficient of kinetic friction (μk) is given as 0.20, and the normal force is 35.28 N.
- Force of friction = 0.20 * 35.28 N = 7.056 N.

3. Calculate the net force acting on the block along the ramp:
- The net force is equal to the component of weight parallel to the ramp minus the force of friction.
- The component of weight acting along the ramp can be calculated as: weight_parallel = weight * sin(angle of incline).
- The angle of incline is given as 30 degrees.
- weight_parallel = 35.28 N * sin(30°) = 17.64 N.
- The net force is: net force = weight_parallel - force of friction.
- net force = 17.64 N - 7.056 N = 10.584 N.

4. Calculate the work done on the block:
- The work done on the block is equal to the product of the net force and the displacement of the block along the ramp.
- The displacement along the ramp is equal to the horizontal distance traveled by the block.
- The horizontal distance can be calculated using the initial speed of the block and the time taken. However, the time is not given.
- Without the time, we cannot calculate the horizontal distance and therefore cannot calculate the work done on the block.

We are unable to calculate the vertical height reached by the block above its starting point without the horizontal distance traveled by the block.

To determine the vertical height the block reaches above its starting point, we need to calculate the work done on the block against gravity. This work is equal to the change in potential energy of the block.

The formula for the work done against gravity is given by:

Work = m * g * h

Where:
m = mass of the block = 3.6 kg
g = acceleration due to gravity = 9.8 m/s^2
h = vertical height

However, to calculate the work, we need to determine the vertical height h first.

To find the height h, we can use the conservation of energy principle. It states that the total mechanical energy of an object is conserved when only conservative forces are doing work.

The initial total mechanical energy of the block includes its kinetic energy and potential energy at the bottom of the ramp:

Ei = KE + PEi

The final total mechanical energy of the block includes its potential energy at the maximum height:

Ef = PEf

According to the conservation of energy principle:

Ei = Ef

KE + PEi = PEf

Since the block starts from rest at the maximum height, its kinetic energy is zero at the top. Therefore, the equation becomes:

PEi = PEf

m * g * h = m * g * h'

Where h' is the height above its starting point.

Now we can solve for h':

m * g * h = m * g * h'

Dividing both sides by m * g:

h = h'

We have found that h = h'. So, the height above its starting point is equal to the vertical height h.

Now, let's solve for h.

Using the equation m * g * h = m * g * h':

3.6 kg * 9.8 m/s^2 * h = 3.6 kg * 9.8 m/s^2 * h'

The mass and acceleration due to gravity cancel out:

h = h'

Therefore, the vertical height the block reaches above its starting point is h. This height can be calculated using the formula:

h = (initial velocity^2 * sin^2(angle)) / (2 * g)

Plugging in the given values:

h = (13 m/s)^2 * sin^2(30¡) / (2 * 9.8 m/s^2)

Calculating the result:

h ≈ 4.71 meters

Therefore, the block reaches a vertical height of approximately 4.71 meters above its starting point.