Let's break down the scenario and analyze what the accelerometer and gyroscope outputs would look like for each axis when the phone is thrown in the air with a rotation (as depicted in the image). --- ## **Assumptions** 1. **Phone is thrown in a parabolic arc and rotates around one axis** (let's assume the y-axis for simplicity). 2. **Coordinate system**: - \(x\): right - \(y\): up - \(z\): out of the screen 3. **Gravity acts downward** (negative y-direction). 4. **Neglect air resistance.** --- ## **Step 1: Motion Analysis** ### **Phases:** 1. **Initial Hold:** Phone is stationary in hand. 2. **Throw:** Phone accelerates upward and forward, then leaves hand. 3. **Free Fall + Rotation:** Phone follows a parabolic arc and rotates. 4. **Landing/Catch:** Phone decelerates rapidly (not shown in the diagram, focus on airborne phase). --- ## **Step 2: Accelerometer Output (\(a_x, a_y, a_z\))** - **While stationary:** - \(a_y \approx -g\) (gravity sensed, downward) - \(a_x, a_z \approx \) - **While airborne (after leaving hand):** - In free fall, **accelerometer senses g on all axes** (only gravity acts, but accelerometer measures non-gravitational acceleration) - \(a_x \approx \), \(a_y \approx \), \(a_z \approx \) - **(Optional: Small variations may occur due to rotation, but the mean is zero)** --- ### **Representative Plots:** - \(a_x\): - **Stationary:** - **Airborne:** - \(a_y\): - **Stationary:** -g - **Airborne:** - \(a_z\): - **Stationary:** - **Airborne:** --- ## **Step 3: Gyroscope Output (\(\Omega_x, \Omega_y, \Omega_z\))** - **While stationary:** - All \(\Omega = \) - **During throw (if rotating around y-axis):** - \(\Omega_y\) will show a nonzero value (angular velocity about y-axis) - \(\Omega_x, \Omega_z \approx \) - **During flight:** - \(\Omega_y\) remains nonzero as the phone continues to spin - \(\Omega_x, \Omega_z \approx \) --- ### **Representative Plots:** - \(\Omega_x\): - **Stationary & Airborne:** - \(\Omega_y\): - **Stationary:** - **Throw & Airborne:** nonzero (constant if rotation rate is steady) - \(\Omega_z\): - **Stationary & Airborne:** --- ## **Step 4: Drawing the Plots** ### **Accelerometer** - **\(a_x\):** Flat line at . - **\(a_y\):** Drops from \(-g\) (in hand) to (airborne), stays at . - **\(a_z\):** Flat line at . ### **Gyroscope** - **\(\Omega_x\):** Flat line at . - **\(\Omega_y\):** Flat at (in hand), jumps up to a constant (during airborne spin). - **\(\Omega_z\):** Flat line at . --- ## **Final Answer (Summary Table):** | | Before (stationary) | Airborne (rotating) | |--------|---------------------|---------------------| | \(a_x\) | | | | \(a_y\) | -g | | | \(a_z\) | | | | \(\Omega_x\) | | | | \(\Omega_y\) | | const (nonzero) | | \(\Omega_z\) | | | --- ### **Key Points:** - **Accelerometer**: Senses gravity when stationary, zero during free fall. - **Gyroscope**: Only nonzero on the axis of rotation (here, y-axis). --- **If you need actual sketches for the plots, let me know!**

# Step-by-Step Solution for Accelerometer and Gyroscope Outputs ## Assumptions 1. **Phone Motion**: The phone is thrown in a parabolic arc, rotating around the y-axis. 2. **Coordinate System**: - \(x\): right - \(y\): up - \(z\): out of the screen 3. **Gravity** acts downward (negative y-direction). 4. **Neglect Air Resistance**. --- ## Step 1: Analyze Motion Phases ### Phases of Motion - **1. Initial Hold**: Phone is stationary in hand. - **2. Throw**: Phone accelerates upward and forward. - **3. Free Fall with Rotation**: Phone follows a parabolic arc while rotating. - **4. Landing**: Phone decelerates rapidly (not analyzed here). --- ## Step 2: Accelerometer Output (\(a_x, a_y, a_z\)) ### During Different Phases 1. **Stationary Phase**: - \(a_y \approx -g\) (gravity sensed, downward) - \(a_x \approx 0\) - \(a_z \approx 0\) 2. **Airborne Phase** (Post-throw): - The phone experiences free fall, so: - \(a_x \approx 0\) (no horizontal acceleration) - \(a_y \approx 0\) (net acceleration due to gravity is counteracted) - \(a_z \approx 0\) (no acceleration in the z-direction) ### Summary of Accelerometer Outputs | Phase | \(a_x\) | \(a_y\) | \(a_z\) | |----------------|----------|----------|----------| | Stationary | \(0\) | \(-g\) | \(0\) | | Airborne | \(0\) | \(0\) | \(0\) | --- ## Step 3: Gyroscope Output (\(\Omega_x, \Omega_y, \Omega_z\)) ### During Different Phases 1. **Stationary Phase**: - \(\Omega_x \approx 0\) - \(\Omega_y \approx 0\) - \(\Omega_z \approx 0\) 2. **Throw Phase** (if rotating around the y-axis): - \(\Omega_x \approx 0\) - \(\Omega_y \) is nonzero (angular velocity about y-axis) - \(\Omega_z \approx 0\) 3. **Airborne Phase**: - \(\Omega_x \approx 0\) - \(\Omega_y\) remains nonzero (constant if rotation rate is steady) - \(\Omega_z \approx 0\) ### Summary of Gyroscope Outputs | Phase | \(\Omega_x\) | \(\Omega_y\) | \(\Omega_z\) | |----------------|---------------|----------------|---------------| | Stationary | \(0\) | \(0\) | \(0\) | | Airborne | \(0\) | nonzero | \(0\) | --- ## Key Points - **Accelerometer**: - Detects gravity when stationary and shows zero acceleration during free fall. - **Gyroscope**: - Shows nonzero angular velocity only on the axis of rotation (y-axis in this case). --- ## Final Summary This analysis provides a clear view of the accelerometer and gyroscope outputs for a phone thrown into the air with rotation.