Final answer:
Using energy conservation and trigonometry, we find the speed of the block at the bottom of the ramp by setting the initial potential energy equal to the final kinetic energy, which allows us to solve for the final speed.
Step-by-step explanation:
To calculate the speed of the block as it reaches the bottom of the ramp, we will use the principles of energy conservation in a frictionless environment. The only forces at work here are gravitational, causing the block to accelerate down the ramp.
The potential energy (PE) at the start will be converted entirely into kinetic energy (KE) at the bottom since there's no friction and no air resistance. The equation for potential energy is PE = mgh, where m is mass, g is the acceleration due to gravity (9.81 m/s²), and h is the height.
To find the height, we use trigonometry: h = L * sin(θ), where L is the length of the ramp and θ is the angle of the incline. Substituting 20.0 m for L and 17.0° for θ, we get h = 20.0 m * sin(17.0°).
The kinetic energy at the bottom is given by KE = 0.5 * m * v², where v is the final velocity we need to find. By setting PE equal to KE, we can solve for v: mgh = 0.5 * m * v². The mass m cancels out, and we can solve for v as v = sqrt(2gh).
After calculating the height and substituting it and g into the equation, we find the speed of the block at the bottom of the ramp.