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The uploaded image contains a problem related to gravitational acceleration involving a ball dropped from a certain height. Here's a breakdown of the given problems and their required solutions:

### Problem Description:
A ball is released from rest above the surface of the Earth, and its position is measured as a function of time. Gravitational acceleration is assumed to be approximately constant near the Earth's surface, and the equations of motion need to be analyzed under this assumption.

### Part (a):
The first part asks to write an expression for the time variable \( t \) of the ball's motion in terms of measurable quantities of distance \( x \) and velocity \( v \). The equation should be expressed in terms of constants.

**Equation:**
The general equation of motion for an object under constant acceleration (gravity, \( g \)) is:
\[
x = x_0 + v_0 t + \frac{1}{2} a t^2
\]
where \( x_0 \) is the initial position, \( v_0 \) is the initial velocity, and \( a \) is the acceleration due to gravity.

Since the ball is released from rest, \( v_0 = 0 \) and \( x_0 = 0 \), so:
\[
x = \frac{1}{2} g t^2
\]
Solving for \( t \):
\[
t = \sqrt{\frac{2x}{g}}
\]

### Part (b):
This part asks to select three measurable variables from the given options (distance \( x \), velocity \( v \), time \( t \), and gravitational acceleration \( g \)). The user needs to specify which of these variables are independent, dependent, or constant.

**Options and Definitions:**
- **Independent Variable:** Usually, time (\( t \)) is considered independent.
- **Dependent Variable:** Position (\( x \)) and velocity (\( v \)) are dependent on time.
- **Constant:** Gravitational acceleration (\( g \)) is constant near the Earth's surface.

### Part (c):
The third part requires calculating the height \( x_0 \) from which the ball was dropped based on the expression derived.

Given the initial conditions where \( v_0 = 0 \), we use:
\[
x_0 = \frac{1}{2} g t^2
\]

### Part (d):
In this section, it is required to derive the expression of velocity \( v \) of the ball as it reaches a specific point.

**Expression for Velocity:**
Using the kinematic equation:
\[
v = v_0 + at
\]
Since \( v_0 = 0 \):
\[
v = gt
\]
Alternatively, using the energy conservation approach:
\[
v = \sqrt{2gx}
\]

Would you like me to provide further details or help with any specific part of these solutions?

### Follow-Up Questions:
1. How do you calculate the time it takes for the ball to hit the ground from a known height?
2. How does changing the height affect the final velocity of the ball?
3. What are the assumptions made in these equations of motion?
4. How would the presence of air resistance alter the equations used in this problem?
5. How do you derive the expression for acceleration from the velocity-time graph?

**Tip:** Always check units to ensure that your equations are consistent, especially when working with physical quantities like distance, time, and acceleration.

A ball is released from rest above the surface of the Earth, and its position is measured as a function of time. Gravitational acceleration is assumed to be approximately constant near the Earth's surface. The questions involve writing an expression for time in terms of distance and velocity, identifying measurable variables, calculating the height from which the ball was dropped, and deriving the expression for velocity.

Explore the equations of motion for a ball dropped from rest under constant gravitational acceleration near the Earth's surface. Learn how to derive expressions for time, velocity, and height, and understand the relationships between these variables. Suitable for high school physics students.

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