## Step-by-Step Solution ### **Step 1: Understand the Scenario** - The image shows a person throwing a smartphone in a parabolic arc. - The phone rotates while in the air (as shown by the dashed curve and the orientation change). - Axes \(x\), \(y\), \(z\) are defined as the phone's local axes. - We are to sketch (qualitatively) the **accelerometer** (\(a_x, a_y, a_z\)) and **gyroscope** (\(\Omega_x, \Omega_y, \Omega_z\)) outputs over time. --- ### **Step 2: State Assumptions** - The phone is thrown such that it rotates mainly about one axis (let’s assume the \(y\)-axis for simplicity). - The phone is in free fall (no contact with hand) after being thrown. - Gravity acts in the negative \(z\)-direction when the phone is upright. - Air resistance is neglected. --- ### **Step 3: Analyze Accelerometer Outputs** - **Before Throw (Held Upright):** - \(a_x \approx \) - \(a_y \approx \) - \(a_z \approx -g\) (gravity, since accelerometer senses the normal force, which cancels gravity when stationary) - **During Free Fall:** - All axes: \(a_x \approx \), \(a_y \approx \), \(a_z \approx \) (Accelerometer measures the net force; in free fall, only gravity acts, so it reads zero—**weightlessness**.) - **After Catch (if caught upright):** - \(a_x \approx \) - \(a_y \approx \) - \(a_z \approx -g\) (back to held position) --- #### **Plots:** - **\(a_x\) and \(a_y\):** Remain around zero throughout. - **\(a_z\):** Starts at \(-g\), drops to \(\) during flight, returns to \(-g\) when caught. --- ### **Step 4: Analyze Gyroscope Outputs** - **Before Throw (Held Upright, Stationary):** - \(\Omega_x = \), \(\Omega_y = \), \(\Omega_z = \) - **During Throw (When Thrown):** - Quick spike in \(\Omega_y\) (if spun about \(y\)-axis) as the phone is given angular velocity. - **During Flight (Rotating, Constant Angular Velocity):** - \(\Omega_y\) stays at a constant value (if only spinning about \(y\)-axis). - \(\Omega_x = \), \(\Omega_z = \). - **After Catch (If stopped):** - \(\Omega_y\) returns to \(\). --- #### **Plots:** - **\(\Omega_x\) and \(\Omega_z\):** Zero throughout (assuming only \(y\)-axis rotation). - **\(\Omega_y\):** Sudden jump to a constant value during the throw, stays constant during flight, drops to zero after catch. --- ### **Step 5: Sketch Representative Plots** #### **Accelerometer:** ``` a_x: |-------------------| (close to zero) a_y: |-------------------| (close to zero) a_z: |---(-g)---| |---(-g)---| (drops to during flight) ``` #### **Gyroscope:** ``` Ω_x: |-------------------| (zero) Ω_y: |------|const|------| (spike to constant, back to zero) Ω_z: |-------------------| (zero) ``` --- ### **Final Answer** #### **Summary Table** | Signal | Before Throw | During Flight | After Catch | |------------|--------------|---------------|--------------| | \(a_x\) | | | | | \(a_y\) | | | | | \(a_z\) | \(-g\) | | \(-g\) | | \(\Omega_x\)| | | | | \(\Omega_y\)| | Constant | | | \(\Omega_z\)| | | | #### **Assumption** - The phone rotates only about the \(y\)-axis after the throw, and air resistance is neglected. --- #### **Explanatory Diagram** - **Accelerometer:** \[ \begin{align*} a_z &: \text{Starts at } -g, \text{ goes to } \text{ during flight, returns to } -g \\ a_x, a_y &: \text{Stay at } \end{align*} \] - **Gyroscope:** \[ \begin{align*} \Omega_y &: \text{ before throw, jumps to constant during flight, after catch} \\ \Omega_x, \Omega_z &: \end{align*} \] --- If you need a hand-drawn sketch of the plots, let me know!