If the height of the water slide in the figure below is h1 = 1.55 m and h2 = 1.35 m, and the person's initial speed at point A is 0.593 m/s, at what location does the swimmer splash down in the pool?
To determine the location where the swimmer splashes down in the pool, we need to apply the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system remains constant if no external forces are acting on it.
At point A, the swimmer's initial mechanical energy can be calculated as the sum of their potential energy and kinetic energy:
Initial Mechanical Energy at A = Potential Energy at A + Kinetic Energy at A
The potential energy of the swimmer at point A is given by:
Potential Energy at A = mass × gravity × height of point A
Here, the mass is not provided in the question, but it cancels out when we compare the potential energy at point A with the potential energy at the final point of splashdown.
The kinetic energy of the swimmer at point A is given by:
Kinetic Energy at A = 0.5 × mass × (initial speed at A)^2
Now, we can equate the initial mechanical energy at point A to the potential energy at the final splashdown point. At the splashdown point, the only remaining potential energy is due to the height of the water slide, and the swimmer's kinetic energy is completely converted into potential energy. Therefore:
Potential Energy at Splashdown = mass × gravity × height of water slide
Since the total mechanical energy is conserved, we can set the initial mechanical energy equal to the potential energy at the splashdown point:
Potential Energy at Splashdown = Initial Mechanical Energy at A
mass × gravity × height of water slide = mass × gravity × height of point A + 0.5 × mass × (initial speed at A)^2
Now, let's solve for the height of the water slide at the splashdown point by rearranging the equation:
height of water slide = (height of point A × gravity + 0.5 × (initial speed at A)^2) / gravity
Substituting the given values:
height of water slide = (1.55 × 9.8 + 0.5 × (0.593)^2) / 9.8
Calculating this expression, the height of the water slide at the splashdown point is:
height of water slide = 1.8995 m
Therefore, the swimmer splashes down in the pool at a height of approximately 1.8995 meters above the ground.