Children slide down a frictionless water slide that ends at a height of 1.80 m above the pool. If a child starts from rest at point A and lands in the water at point B, a horizontal distance L = 2.43 m from the base of the slide, determine the height h of the water slide.
To determine the height h of the water slide, we can use the principle of conservation of mechanical energy. The total mechanical energy at the top of the slide (point A) must be equal to the total mechanical energy at the bottom (point B), neglecting any energy loss due to friction.
The total mechanical energy at the top of the slide consists of two components: potential energy and kinetic energy. The potential energy is given by the formula PE = mgh, where m is the mass of the child, g is the acceleration due to gravity, and h is the height of the slide.
The kinetic energy at the top of the slide is zero as the child starts from rest. Therefore, the total mechanical energy at point A is equal to mgh.
At the bottom of the slide (point B), the child has only kinetic energy since the height is zero. The kinetic energy is given by the formula KE = 1/2mv^2, where v is the velocity of the child at point B.
Since we know the horizontal distance L from the base of the slide and the height difference between A and B, we can use the following equation to relate the velocities at A and B:
v^2 = v_A^2 + 2gL
Since the slide is frictionless, the velocity at point A is zero. Therefore, the equation becomes:
v^2 = 0 + 2gL
Simplifying, we have:
v^2 = 2gL
Now, we can equate the total mechanical energy at point A with the kinetic energy at point B and solve for the height h:
mgh = 1/2mv^2
Simplifying and canceling out the mass m, we have:
gh = 1/2v^2
Substituting the value of v^2 from earlier, we have:
gh = 1/2(2gL)
Simplifying further, we get:
gh = gL
Finally, we can cancel out the acceleration due to gravity, g, to solve for the height h:
h = L
Substituting the given value for L, we find:
h = 2.43 m
Therefore, the height of the water slide is 2.43 meters.
To determine the height h of the water slide, we can use the conservation of energy principle. The initial potential energy at point A is converted into kinetic energy at point B.
Let's assume the mass of the child is m, the acceleration due to gravity is g, and the speed of the child at point B is v.
1. At the starting point A:
- Potential Energy (PE) = mgh (where h is the height of the slide)
- Kinetic Energy (KE) = 0 (the child starts from rest, so no initial kinetic energy)
2. At the ending point B:
- Potential Energy (PE) = 0 (the child lands in the water at point B)
- Kinetic Energy (KE) = (1/2)mv^2
According to the conservation of energy principle, the initial potential energy equals the final kinetic energy:
mgh = (1/2)mv^2
The mass m cancels out, and we are left with:
gh = (1/2)v^2
Solving for h, we get:
h = (1/2)v^2 / g
We need to determine the value of v. As the slide is frictionless, the child will experience no energy loss due to friction. The vertical height lost by the child is the same as the horizontal distance traveled. Therefore, we can use the range equation:
L = v√(2h/g)
Simplifying for v^2, we have:
v^2 = (L^2 * g) / (2h)
Substituting this into the previous equation, we have:
h = (1/2) * ((L^2 * g) / (2h)) / g
Simplifying further:
h = (L^2) / (4h)
Multiplying both sides by 4h:
4h^2 = L^2
Simplifying and solving for h:
h^2 = L^2 / 4
h = √(L^2 / 4)
Plugging in the given value of L = 2.43 m:
h = √(2.43^2 / 4)
h = √(5.9049 / 4)
h = √1.476225
h ≈ 1.214 m
Therefore, the height h of the water slide is approximately 1.214 meters.
how long does it take to fall 1.8m?
1.8=1/2 g t^2 solve for time t.
then velocity horizontal=2.43/time