4. A 500-g block is shot up a surface 25° inclined with the horizontal with an initial speed of 200cm/s. How far up the incline will it go if the coefficient of friction between it and the incline is 0.150?

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To find out how far up the incline the block will go, we can use the concept of work and energy. Here are the steps to solve the problem:

Step 1: Determine the gravitational force acting on the block.
The gravitational force can be calculated using the formula:
F_gravity = mass × acceleration due to gravity
=> F_gravity = 500 g × 9.8 m/s^2

Step 2: Calculate the force of friction.
The force of friction can be determined using the formula:
F_friction = coefficient of friction × F_normal
Where F_normal is the normal force acting on the block perpendicular to the surface of the incline. Since the incline is at an angle of 25° with the horizontal, the normal force can be calculated as:
F_normal = F_gravity × cos(25°)

Step 3: Calculate the net force and acceleration of the block.
The net force acting on the block can be determined by subtracting the force of friction from the component of the gravitational force parallel to the incline:
F_net = F_gravity × sin(25°) - F_friction

The net force can also be calculated using Newton's second law:
F_net = mass × acceleration

Step 4: Determine the distance traveled up the incline.
To find the distance traveled up the incline, we can use the work-energy principle. The work done on the block is equal to the change in its kinetic energy. Therefore:
Work_done = change in kinetic energy
=> Force × distance × cos(0°) = (1/2) m (vf^2 - vi^2)
Here, vf represents the final velocity (which is zero at the highest point), vi represents the initial velocity (given as 200 cm/s), and m is the mass of the block. Rearrange the equation to solve for distance:
Distance = (vf^2 - vi^2) / (2 × acceleration)

By substituting the values and solving the equation, we can find the distance up the incline the block will go.