If the height of the water slide in the figure is h = 2.8 m and the person's initial speed at point A is 0.47 m/s, what is the new horizontal distance L between the base of the slide and the splashdown point of the person?

To find the new horizontal distance L between the base of the slide and the splashdown point of the person, we can use the conservation of energy.

1. First, let's determine the potential energy at point A. The potential energy at any given point is given by the formula PE = m * g * h, where m is the mass of the person, g is the acceleration due to gravity, and h is the height.

2. Since the problem does not provide the mass of the person, we can assume a standardized mass, such as 1 kg, for simplicity. So the potential energy at point A is PE_A = 1 kg * 9.8 m/s^2 * 2.8 m.

3. Next, let's consider the kinetic energy at point A. The initial speed at point A is given as 0.47 m/s. The kinetic energy at any given point is given by the formula KE = (1/2) * m * v^2, where v is the velocity.

4. Plugging in the given values, the kinetic energy at point A is KE_A = (1/2) * 1 kg * (0.47 m/s)^2.

5. According to the conservation of energy, the change in potential energy should be equal to the change in kinetic energy. In this case, as the person slides down the water slide, the potential energy decreases and the kinetic energy increases.

6. Therefore, we can set up the equation PE_A - PE_splash = KE_splash - KE_A, where PE_splash is the potential energy at the splashdown point, and KE_splash is the kinetic energy at the splashdown point.

7. Since the person comes to a stop at the splashdown point, the kinetic energy at the splashdown point, KE_splash, is equal to 0.

8. We can rewrite the equation as PE_A - PE_splash = 0 - KE_A.

9. Now, let's solve for PE_splash by rearranging the equation: PE_splash = PE_A - KE_A.

10. Substitute the values into the equation: PE_splash = (1 kg * 9.8 m/s^2 * 2.8 m) - [(1/2) * 1 kg * (0.47 m/s)^2].

11. Calculate the potential energy at the splashdown point, PE_splash.

12. Once we have the potential energy at the splashdown point, we can use it to find the new horizontal distance L using the equation PE_splash = m * g * L, where L is the horizontal distance.

13. Rearrange the equation to solve for L: L = PE_splash / (m * g).

14. Substitute the values into the equation: L = PE_splash / (1 kg * 9.8 m/s^2).

15. Calculate the new horizontal distance L.

To find the new horizontal distance (L) between the base of the slide and the splashdown point of the person, we need to use the principles of conservation of energy and kinematics.

First, let's break down the problem into steps:

Step 1: Calculate the vertical height (h) of the water slide.
Step 2: Determine the speed of the person at the bottom of the slide using conservation of energy.
Step 3: Calculate the time it takes for the person to reach the bottom using kinematics.
Step 4: Use the horizontal velocity and time to find the horizontal distance (L) traveled.

Step 1: Calculate the vertical height (h) of the water slide.
Given: h = 2.8 m

Step 2: Determine the speed of the person at the bottom of the slide using conservation of energy.
The conservation of energy tells us that the potential energy at the top (U) is equal to the kinetic energy at the bottom (K). We can write this as:

mgh = (1/2)mv^2

Where:
m = mass of the person (which we will assume to be canceled out as it appears on both sides of the equation)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height of the slide = 2.8 m
v = speed of the person at the bottom of the slide (which we need to find)

Simplifying the equation, we get:

gh = (1/2)v^2

Solving for v, we find:

v = √(2gh)

Substituting the given values, we get:

v = √(2 * 9.8 * 2.8)

Step 3: Calculate the time it takes for the person to reach the bottom using kinematics.
We can use the kinematic equation to calculate the time it takes for the person to fall from height h. The equation is:

h = (1/2)gt^2

Where:
h = height of the slide = 2.8 m
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time taken to reach the bottom (which we need to find)

Rearranging the equation to solve for t, we get:

t = √(2h/g)

Substituting the given values, we get:

t = √(2 * 2.8 / 9.8)

Step 4: Use the horizontal velocity and time to find the horizontal distance (L) traveled.
Since the person's initial speed at point A is given as 0.47 m/s, we assume that the horizontal velocity remains constant throughout the slide.

L = v * t

Substituting the values of v and t that we calculated earlier, we can find the horizontal distance, L.