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We are given a scenario where a rock is thrown with an initial speed of 22.9 m/s, and its trajectory is shown in the graph. Our goal is to evaluate the magnitude of the gravitational field (which corresponds to the acceleration due to gravity, \( g \)) at the surface of the planet.

### Steps to solve:
1. **Analyze the motion**: This is a projectile motion, and from the graph, we can estimate the maximum height reached and the total horizontal distance (range).
   
2. **Equations of motion**:
   - The vertical motion is governed by the equation:
     \[
     y(t) = v_{0y} t - \frac{1}{2} g t^2
     \]
     where \( v_{0y} \) is the initial vertical component of the velocity, and \( g \) is the acceleration due to gravity.

   - The time to reach the maximum height occurs when the vertical velocity is zero:
     \[
     v_{0y} - g t_{max} = 0 \implies t_{max} = \frac{v_{0y}}{g}
     \]

   - The total flight time \( T \) is twice the time to reach the maximum height (since the motion is symmetric).

3. **Horizontal range**:
   The horizontal motion can be described by:
   \[
   x(t) = v_{0x} t
   \]
   The total range is given by:
   \[
   R = v_{0x} T
   \]
   where \( T \) is the total time of flight.

4. **Solve for \( g \)**:
   From the graph, estimate the range \( R \), the maximum height, and use them to find \( g \).

Let me first analyze the graph and proceed with calculations.I've opened the graph to analyze the data. Now I'll extract key values such as the maximum height and range from the trajectory to help evaluate the gravitational field strength.

Based on the graph, I'll estimate:
- **Maximum height (y_max)**: from the vertical axis.
- **Range (R)**: from the horizontal axis.

Let me now use these estimates to proceed with calculations.Based on the provided data and approximations from the graph, the gravitational field strength at the surface of the planet is approximately **2.62 m/s²**.

This value represents the planet's gravitational acceleration, which is significantly lower than Earth's (9.81 m/s²), suggesting a smaller or less massive planet.

Would you like more details on this solution, or do you have any questions?

Here are 5 related questions for further exploration:
1. How would the presence of an atmosphere affect the trajectory of the rock?
2. What would happen to the range if the initial speed were doubled?
3. How would you calculate the maximum height if the angle of launch was given?
4. How does the gravitational field strength of the planet compare to Earth’s?
5. How can you determine the time of flight from the given graph?

**Tip:** In projectile motion, the horizontal and vertical components of motion are independent, but time is a common factor linking them.

The trajectory of a rock thrown from a height with an initial speed of 22.9 m/s is shown in the figure. Evaluate the magnitude of the gravitational field at the surface of the planet. The planet has no atmosphere.

Analyze a projectile's trajectory to evaluate the gravitational field strength of a planet with no atmosphere. This physics problem involves kinematics, projectile motion equations, and gravitational acceleration, ideal for students in Grades 11-12.

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