Final answer:
Using the conservation of energy, the height of the inclined ramp, and the lack of friction and air resistance, the speed of the block at the bottom of a 20-meter inclined ramp at a 17-degree angle to the horizontal is approximately 10.6 m/s.
Step-by-step explanation:
To calculate the speed of the block as it reaches the bottom of the frictionless inclined ramp, we can use the principles of conservation of energy. Since there is no resistance or friction, all the potential energy at the top of the ramp will be converted into kinetic energy at the bottom.
The potential energy (PE) at the top is given by PE = mgh, where m is the mass of the block, g is the acceleration due to gravity (9.8 m/s2), and h is the height of the ramp. We can find the height by using trigonometry, knowing the length of the ramp (20 m) and the angle of inclination (17 degrees).
The height h can be calculated as h = L sin(θ), where L is the length of the ramp and θ is the angle with the horizontal. Once we know the height, we can calculate the potential energy. Since the block is starting from rest, the kinetic energy (1/2 mv2) at the top is zero. At the bottom of the ramp, the potential energy will be zero, and all the energy will be kinetic.
Setting the potential energy equal to the kinetic energy, we get:
mgh = 1/2 mv2
We can cancel m from both sides and solve for v:
gh = 1/2 v2
v2 = 2gh
v = sqrt(2gh)
nHeight h can be found by:
h = 20 sin(17°) = 5.7 m approximately.
Then substitute h into the previous equation to find v:
v = sqrt(2 * 9.8 * 5.7) = sqrt(111.72)
The speed of the block at the bottom of the ramp is approximately 10.6 m/s.