Using GRESA method, solve the problem: With what speed in miles/hr (1 m/s = 2.23 mi/hr) must an object be thrown to reach a height of 91.5 m (equivalent to one football field)? Assume negligible air resistance.

To solve this problem using the GRESA method (Given, Required, Equation, Solution, Answer), we can break down the steps as follows:

### 1. **Given**: 
- The height the object needs to reach: \( h = 91.5 \) meters.
- Conversion factor: \( 1 \) m/s = \( 2.23 \) miles/hour.
- Acceleration due to gravity: \( g = 9.8 \) m/s\(^2\) (since the object is thrown upward, gravity will act in the opposite direction).

### 2. **Required**: 
- The initial speed \( v_0 \) in miles per hour needed for the object to reach a height of 91.5 meters.

### 3. **Equation**: 

To find the initial speed required to reach a certain height, we can use the kinematic equation for motion under constant acceleration:

\[
v^2 = v_0^2 + 2a s
\]

Since the object reaches maximum height, the final velocity \( v = 0 \). Also, \( a = -g \) because gravity acts downwards, and \( s = h = 91.5 \) meters:

\[
0 = v_0^2 - 2gh
\]

Rearrange this equation to solve for \( v_0 \):

\[
v_0^2 = 2gh
\]

\[
v_0 = \sqrt{2gh}
\]

### 4. **Solution**:

Substitute the given values into the equation:

\[
v_0 = \sqrt{2 \cdot 9.8 \, \text{m/s}^2 \cdot 91.5 \, \text{m}}
\]

\[
v_0 = \sqrt{1791}
\]

\[
v_0 \approx 42.33 \, \text{m/s}
\]

Now, convert this speed to miles per hour:

\[
v_0 \approx 42.33 \, \text{m/s} \times 2.23 \, \frac{\text{mi/hr}}{\text{m/s}}
\]

\[
v_0 \approx 94.40 \, \text{mi/hr}
\]

### 5. **Answer**: 

The object must be thrown with an initial speed of approximately **94.40 miles per hour** to reach a height of 91.5 meters, assuming negligible air resistance.

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Do you want more details or have any questions?  

### 5 Related Questions:

1. What would the initial speed be if the object had to reach a height of 50 meters?
2. How would the initial speed change if the object's height to be reached was doubled?
3. What would happen to the initial speed if the object is thrown on a planet with a different gravitational acceleration?
4. How much time would it take for the object to reach its maximum height?
5. How would air resistance affect the initial speed required to reach 91.5 meters?

### Tip:

When solving physics problems, always keep track of your units. Make sure to convert units when necessary to maintain consistency throughout your calculations.

Learn how to calculate the initial speed required for an object to reach a height of 91.5 meters using the kinematic equation under constant acceleration. Includes step-by-step solution and conversion of units from meters per second to miles per hour.

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