A 2.00kg block is pushed against a spring with force constant k=400N(m^-1), compressing it 0.180m. The block is then released and moves along a frictionless horizontal surface and then up frictionless incline with slope angle 37.0degrees.

a. What is the speed of the block as it slides along the horizontal surface after having left the spring?
b. How far does the block travel up the incline before starting to slide back down.

A) Speed(v)=2.55m/s B) S=0.54m

To solve these problems, we need to apply the principles of conservation of mechanical energy and Newton's second law.

a. To find the speed of the block as it slides along the horizontal surface after leaving the spring, we can use the principle of conservation of mechanical energy. The initial potential energy stored in the spring is converted into kinetic energy as the block moves along the horizontal surface.

1. Calculate the potential energy stored in the spring when it is compressed by 0.180m:
Potential Energy = (1/2)kx^2
where k is the force constant and x is the displacement from the equilibrium position.
Potential Energy = (1/2) * 400 N/m * (0.180 m)^2

2. The potential energy is converted into kinetic energy as the block moves along the horizontal surface. At this point, the potential energy is zero. So, we can equate the initial potential energy to the final kinetic energy:
Potential Energy = Kinetic Energy
(1/2) * 400 N/m * (0.180 m)^2 = (1/2) * m * v^2
where m is the mass of the block and v is the final velocity.

3. Rearranging the equation and solving for v:
v^2 = (400 N/m * (0.180 m)^2) / 2.00 kg
v = sqrt((400 N/m * (0.180 m)^2) / 2.00 kg)

Therefore, the speed of the block as it slides along the horizontal surface after leaving the spring can be calculated using the formula mentioned above.

b. To find how far the block travels up the incline before starting to slide back down, we need to find the point where the net force acting on the block changes signs. At this point, all the potential energy is converted into kinetic energy. The sum of the gravitational potential energy and the potential energy stored in the spring equals the kinetic energy of the block.

1. Calculate the gravitational potential energy at the highest point on the incline:
Gravitational Potential Energy = m * g * h
where m is the mass of the block, g is the acceleration due to gravity, and h is the vertical height.

2. The potential energy stored in the spring is released as the block moves up the incline. At the highest point, all the potential energy is converted into kinetic energy. So, we can equate the combined potential energy to the kinetic energy:
(1/2) * k * x^2 + m * g * h = (1/2) * m * v^2
where k is the force constant, x is the displacement of the spring, and v is the velocity of the block.

3. Rearranging the equation and solving for x:
x = sqrt((m * g * h) / k)

Therefore, the distance traveled by the block up the incline before starting to slide back down can be calculated using the formula mentioned above.