The drinking fountain is designed such that the nozzle is located from the edge of the basin as shown. Suppose that θ = 26 ∘ a) Determine the maximum speed at which water can be ejected from the nozzle so that it does not splash over the sides of the basin at  B and  C. b) Determine the minimum speed at which water can be ejected from the nozzle so that it does not splash over the sides of the basin at  B
  and  C
 .


To solve this problem, we can break it down into a projectile motion problem, where water is ejected at an angle from the nozzle and we need to determine the speed that ensures the water falls within the basin.

### Assumptions and Variables:
1. **Nozzle angle:** \( \theta = 26^\circ \)
2. **Horizontal distances:** 
   - The nozzle is located at a certain horizontal distance from points B and C on the basin.
   - These distances need to be known or given in the diagram for solving the problem (if unspecified, they are essential for solving).

3. **Projectile motion equations:**
   - The horizontal motion is governed by \( x = v_0 \cos(\theta) t \)
   - The vertical motion is governed by \( y = v_0 \sin(\theta) t - \frac{1}{2} g t^2 \)

Where:
- \( v_0 \) = initial velocity (speed) of water from the nozzle,
- \( \theta = 26^\circ \) (angle of projection),
- \( t \) = time,
- \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity),
- \( x \) = horizontal distance from the nozzle,
- \( y \) = vertical distance from the nozzle to a point (like B or C).

### **Part (a) Maximum speed:**

We need to find the maximum speed at which water can be ejected so that it doesn’t overshoot the basin, meaning it must land between points B and C. Let’s denote:
- \( x_B \) = horizontal distance to point B,
- \( x_C \) = horizontal distance to point C.

The water should land before reaching point C (the farthest point from the nozzle).

#### Solving the projectile motion:
At the point where the water reaches point B or C, the vertical displacement is determined by the depth of the basin, say \( y_B \) (for point B) and \( y_C \) (for point C).

Using the horizontal and vertical projectile motion equations, the relationship between time and initial speed can be obtained as:

- Horizontal motion: \( x = v_0 \cos(\theta) t \)
- Vertical motion: \( y = v_0 \sin(\theta) t - \frac{1}{2} g t^2 \)

To prevent water from splashing beyond point C, solve these equations to find the maximum \( v_0 \).

### **Part (b) Minimum speed:**

For minimum speed, the water should reach at least point B without falling short. Thus, at point B, solve for the initial speed \( v_0 \) that ensures the water lands exactly at point B.

#### Steps to solve:
1. **For maximum speed:** Use the projectile motion equations to ensure the water reaches exactly at point C.
2. **For minimum speed:** Use the same equations, but solve to ensure the water lands at point B.

### Next Steps:

- Could you provide more information or a diagram with the dimensions of the basin, including the horizontal distances to points B and C, as well as the height difference between the nozzle and the basin bottom? These are necessary for solving the problem precisely.

Would you like more details on how to apply the projectile motion equations, or do you have specific values I should use for the dimensions of the basin?

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### Relative Questions:
1. What are the horizontal distances to points B and C in this scenario?
2. Is there a height difference between the nozzle and the bottom of the basin?
3. Should we consider air resistance in this problem, or ignore it for simplicity?
4. What are typical values for water ejection speed from fountains, if known?
5. How does the shape of the basin influence the solution, if at all?

### Tip:
In projectile motion problems, always separate the horizontal and vertical motions and use the time of flight to connect them for solving.

The drinking fountain is designed such that the nozzle is located from the edge of the basin as shown. Suppose that θ = 26 ∘ a) Determine the maximum speed at which water can be ejected from the nozzle so that it does not splash over the sides of the basin at B and C. b) Determine the minimum speed at which water can be ejected from the nozzle so that it does not splash over the sides of the basin at B and C.

Solve the projectile motion problem of determining the maximum and minimum speed at which water can be ejected from a nozzle at a 26-degree angle to prevent splashing at points B and C in a fountain basin. This solution covers kinematic equations and trigonometric applications, suitable for students in Grades 9-12.

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