https://r2.erweima.ai/mathbot/image/1726933880034roui8z11.jpg

The problem you have shared involves determining the initial minimum velocity \( v_{0MIN} \) and the maximum velocity \( v_{0MAX} \) of defective grains that fall into a deposit. From the image, the provided data and the formulas for both velocities involve projectile motion equations.

### Given Data:
- \( \Delta x_{\text{min}} = 1.25 \, \text{m} \)
- \( \Delta x_{\text{max}} = 1.85 \, \text{m} \)
- \( \Delta y = 0.15 \, \text{m} \)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( \theta_0 = 18^\circ \) (launch angle)

### Equations Used:
- For the **minimum velocity**:
  \[
  \Delta y = \Delta x \cdot \tan(\theta_0) - \frac{g \cdot \Delta x^2}{2 v_0^2 \cos^2(\theta_0)}
  \]
  Applying this equation with the minimum horizontal distance \( \Delta x = 1.25 \, \text{m} \), you get:
  \[
  v_{0MIN} = 5.74 \, \text{m/s}
  \]

- For the **maximum velocity**:
  The same equation applies, but with the maximum horizontal distance \( \Delta x = 1.85 \, \text{m} \), leading to:
  \[
  v_{0MAX} = 6.41 \, \text{m/s}
  \]

### Summary:
The calculations give:
- **Minimum initial velocity** \( v_{0MIN} = 5.74 \, \text{m/s} \)
- **Maximum initial velocity** \( v_{0MAX} = 6.41 \, \text{m/s} \)

Do you need any further explanations or detailed steps for the calculations?

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Here are some follow-up questions for practice:
1. How does the launch angle affect the minimum and maximum velocities?
2. What would happen to the velocities if the gravitational constant changed?
3. How would increasing the height \( \Delta y \) affect the initial velocities?
4. Can the range of velocities be changed by altering only the horizontal distances?
5. What if the launch angle was instead 45 degrees? How would the results differ?

**Tip**: In projectile motion problems, angles near 45 degrees maximize the range for a given velocity!

Determine the initial minimum velocity v_0MIN and the maximum velocity v_0MAX of defective grains that fall into the deposit.

Solve a physics problem involving projectile motion to determine the initial minimum and maximum velocities of defective grains falling into a deposit. Given values include Δx_min = 1.25 m, Δx_max = 1.85 m, Δy = 0.15 m, g = 9.8 m/s^2, and tan(θ_0) = 18°. Learn how to apply kinematic equations to calculate v_0MIN = 5.74 m/s and v_0MAX = 6.41 m/s.

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