Lezioni di Fisica Generale I

It seems like you are trying to describe the motion of an object under constant acceleration, likely in a vertical direction (e.g., free fall or projectile motion), and then solve for specific points in time. Here is a cleaned-up version with some corrections and explanations: ### Given: - \( \vec{g} = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity) - Initial velocity: \( v_0 \) - Initial position: \( s_0 \) The equations of motion for an object under constant acceleration are: 1. **Position as a function of time**: \[ s(t) = s_0 + v_0 t - \frac{1}{2} g t^2 \] 2. **Velocity as a function of time**: \[ v(t) = v_0 - g t \] 3. **Acceleration is constant and equal to \( g \)**. ### Step-by-Step Solution: 1. **Initial Conditions**: At \( t = 0 \): \[ s(0) = s_0, \quad v(0) = v_0 \] 2. **Finding the Time When Velocity is Zero**: Set \( v(t) = 0 \): \[ 0 = v_0 - g t \implies t = \frac{v_0}{g} \] 3. **Position at this time**: Substitute \( t = \frac{v_0}{g} \) into the position equation: \[ s\left(\frac{v_0}{g}\right) = s_0 + v_0 \cdot \frac{v_0}{g} - \frac{1}{2} g \left(\frac{v_0}{g}\right)^2 \] Simplify: \[ s\left(\frac{v_0}{g}\right) = s_0 + \frac{v_0^2}{g} - \frac{1}{2} \cdot \frac{v_0^2}{g} \] \[ s\left(\frac{v_0}{g}\right) = s_0 + \frac{v_0^2}{2g} \] 4. **Finding the Maximum Height**: The maximum height is reached when \( v(t) = 0 \). So, at this point: \[ h_{\text{max}} = s\left(\frac{v_0}{g}\right) = s_0 + \frac{v_0^2}{2g} \] 5. **Finding the Time When Position is Zero** (assuming \( s_0 = 0 \)): Set \( s(t) = 0 \): \[ 0 = v_0 t - \frac{1}{2} g t^2 \] Factor out \( t \): \[ t \left( v_0 - \frac{1}{2} g t \right) = 0 \] So, the solutions are: \[ t = 0 \quad \text{(initial time)} \quad \text{or} \quad t = \frac{2v_0}{g} \] 6. **Position at \( t = \frac{2v_0}{g} \)** (time when the object returns to the ground): \[ s\left(\frac{2v_0}{g}\right) = v_0 \cdot \frac{2v_0}{g} - \frac{1}{2} g \left(\frac{2v_0}{g}\right)^2 \] Simplify: \[ s\left(\frac{2v_0}{g}\right) = \frac{2v_0^2}{g} - 2v_0^2 / g = 0 \] ### Summ