A water slide is constructed so that swimmers, starting from rest at the top of the slide, leave the end of the slide traveling horizontally. As the drawing shows, one person hits the water 5.00 m from the end of the slide in a time of 0.386 s after leaving the slide. Ignore friction and air resistance, find the height H.

To find the height H, we can use the equations of motion. Let's assume that the initial velocity of the swimmer at the top of the slide is zero.

First, we need to find the horizontal velocity (Vx) at the point where the swimmer hits the water. We can use the equation:

Vx = (displacement) / (time)
Vx = 5.00 m / 0.386 s
Vx ≈ 12.95 m/s

Since there is no horizontal acceleration, the horizontal velocity remains constant throughout the motion.

Now, we can find the time it takes for the swimmer to reach the water from the top of the slide. We can use the equation:

Time = (displacement) / (initial vertical velocity)
0.386 s = (H) / (V0)

Since the initial vertical velocity (V0) is zero, we can ignore it in this equation.

Now, rearranging the equation to solve for H:

H = Time * V0
H = 0.386 s * V0

Now, to find V0, we can use the equation that describes motion in the vertical direction:

Vf^2 = Vi^2 + 2aΔy

Since the swimmer starts from rest, the initial vertical velocity (Vi) is zero. The final vertical velocity (Vf) can be derived from the equation:

Vf = g * t

Where g is the acceleration due to gravity (approximately 9.8 m/s²) and t is the time taken to reach the water.

Now, we can substitute the values and solve for Vf:

Vf = 9.8 m/s² * 0.386 s
Vf ≈ 3.787 m/s

Substituting this value back into the equation for H:

H = 0.386 s * V0
H = 0.386 s * 3.787 m/s
H ≈ 1.47 m

Therefore, the height of the water slide (H) is approximately 1.47 meters.