https://r2.erweima.ai/mathbot/image/1724668814312dxkxec1q.jpg

The problem shown in the image involves calculating the time it takes for an object dropped from a certain height to reach the ground. The object starts with an initial downward velocity and is influenced by gravity.

### Problem Breakdown:
1. **Given Information:**
   - Initial velocity \( v_i = 8.1 \, \text{m/s} \) (downward)
   - Distance (height) \( s = 180 \, \text{m} \)
   - The equation for the distance fallen is given as:
     \[
     s = 0.81t + 5t^2
     \]
   where \( t \) is the time in seconds.

2. **Required:**
   - Find the time \( t \) when the object reaches the ground (when \( s = 180 \, \text{m} \)).

### Solving the Equation:
Substitute \( s = 180 \) into the equation:
\[
180 = 0.81t + 5t^2
\]
This is a quadratic equation in the form:
\[
5t^2 + 0.81t - 180 = 0
\]

### Solving for \( t \) using the quadratic formula:
The quadratic formula is:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( 5t^2 + 0.81t - 180 = 0 \):
- \( a = 5 \)
- \( b = 0.81 \)
- \( c = -180 \)

Now, let's calculate the value of \( t \).The quadratic equation yields two solutions for \( t \):

- \( t_1 = 5.92 \, \text{seconds} \)
- \( t_2 = -6.08 \, \text{seconds} \) (not physically meaningful since time cannot be negative)

Therefore, the time it takes for the object to reach the ground is approximately **5.92 seconds**.

Would you like further details or have any other questions?

Here are five related questions you might find interesting:
1. How would the result change if the initial velocity were doubled?
2. What happens if the object is thrown upwards instead of downwards?
3. How does air resistance affect the time of descent?
4. Can this method be used for objects dropped from different altitudes?
5. What if the initial velocity is zero? How does that affect the calculation?

**Tip:** When solving quadratic equations in physics, always check the physical relevance of the roots—negative time, for instance, is not meaningful in most real-world scenarios.

Learn how to calculate the time it takes for an object to reach the ground using quadratic equations in physics. This problem involves kinematics and gravity, with a detailed step-by-step solution. Suitable for high school students.

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