A small package is deposited by the conveyor belt onto the 30o ramp at A with a velocity of0.8m/s. Calculate the distance s on the level surface BC at which the package comes to rest.The coefficient of kinetic friction for the package and supporting surface from A to C is 0.3.

To solve this problem, we need to use the principles of physics, specifically the laws of motion and the concept of friction. Here are the steps:

1. First, we need to calculate the acceleration of the package on the ramp. The forces acting on the package are gravity and friction. The force due to gravity is mgsin(30) (where m is the mass of the package, g is the acceleration due to gravity, and 30 is the angle of the ramp). The force due to friction is μmg*cos(30) (where μ is the coefficient of friction). The net force on the package is the difference between these two forces, which gives us the acceleration (a = F/m).

2. Next, we need to find the velocity of the package when it reaches the bottom of the ramp. We can use the equation v^2 = u^2 + 2as (where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance). In this case, the initial velocity is 0.8 m/s, the acceleration is what we calculated in step 1, and the distance is the length of the ramp.

3. Once we have the velocity at the bottom of the ramp, we can calculate the distance the package travels on the level surface BC. The package will come to rest when all its kinetic energy has been converted into work done against friction. We can use the equation KE = W (where KE is the kinetic energy and W is the work done). The kinetic energy is 0.5mv^2 (where v is the velocity at the bottom of the ramp) and the work done is F*d (where F is the force of friction and d is the distance). Solving this equation for d gives us the distance the package travels on the level surface BC.

Note: The mass of the package (m) cancels out in these calculations, so we don't need to know it to solve this problem.

To solve this problem, we need to use the principles of physics, specifically the laws of motion and the concept of friction. Here are the steps:

1. First, we need to calculate the acceleration of the package on the ramp. The acceleration (a) is given by the equation a = g(sinθ - μkcosθ), where g is the acceleration due to gravity (9.8 m/s²), θ is the angle of the ramp (30 degrees), and μk is the coefficient of kinetic friction (0.3).

2. Convert the angle to radians: θ = 30 * π/180 = π/6 rad.

3. Substitute the values into the equation: a = 9.8(sin(π/6) - 0.3cos(π/6)) = 9.8(0.5 - 0.3*0.866) = 9.8(0.5 - 0.2598) = 9.8 * 0.2402 = 2.354 m/s².

4. Now, we need to find the time it takes for the package to come to rest. We can use the equation v = u + at, where v is the final velocity (0 m/s, because the package comes to rest), u is the initial velocity (0.8 m/s), a is the acceleration we just calculated (-2.354 m/s², because it's decelerating), and t is the time. Rearranging the equation gives us t = (v - u) / a = (0 - 0.8) / -2.354 = 0.34 s.

5. Finally, we can find the distance s on the level surface BC at which the package comes to rest. We can use the equation s = ut + 0.5at². Substituting the values gives us s = 0.8 * 0.34 + 0.5 * -2.354 * (0.34)² = 0.272 - 0.136 = 0.136 m.

So, the package comes to rest at a distance of 0.136 m on the level surface BC.