# Step-by-Step Solution ## Problem Restatement We have a rocket flying **horizontally**: - At time \( t \), **mass** = \( M \) - **Velocity** = \( U \) (relative to ground) - **Exhaust gas** comes out at speed \( V_ \) (relative to rocket, i.e., nozzle frame) - **Mass flow rate** of exhaust = \( \dot{m} \) - **Air resistance** = \( D \) (constant) - **Rocket acceleration** = \( a \) - **Find:** Derive an expression for \( a \) (rocket's acceleration) We are told to **choose control volume properly** and explain. --- ## Step 1: Choose Control Volume (C.V.) Choose the **rocket** as the control volume. - It is a **non-deforming, moving C.V.** (the mass changes, but you track the rocket itself, not the surrounding air). - We use the **momentum balance** for a moving, variable-mass system. --- ## Step 2: Apply Linear Momentum Equation (Rocket Equation) General momentum equation for control volume (CV): \[ \sum F_{\text{ext}} = \frac{d}{dt}(M U) - \dot{m}_{\text{out}} (U_{\text{out,rel}}) + \dot{m}_{\text{in}} (U_{\text{in,rel}}) \] Here, mass only **leaves** (no mass enters): - \( U \): velocity of rocket (CV) relative to ground - Exhaust leaves at \( V_ \) **relative to rocket**, so relative to ground: \( U_{\text{exhaust, ground}} = U - V_ \) (If rocket moves at \( U \), and exhaust is ejected backward at \( V_ \) relative to rocket, its ground speed is \( U - V_ \)) So, mass leaves at rate \( \dot{m} \) with velocity \( U - V_ \) (relative to ground). --- ## Step 3: Write the Momentum Balance \[ \sum F_{\text{ext}} = \frac{d}{dt}(MU) = M \frac{dU}{dt} + U \frac{dM}{dt} \] The external forces are: - Drag \( -D \) (opposes motion) - Thrust from exhaust (already accounted for by momentum flux) - (No gravity in horizontal motion) Now, account for **momentum flux**: \[ \text{Net outflow:} \quad -\dot{m}(U_{\text{exhaust, ground}}) = -\dot{m}(U - V_) \] (The negative sign because mass is leaving the rocket.) So, \[ \sum F_{\text{ext}} = M \frac{dU}{dt} + U \frac{dM}{dt} = -D - \dot{m}(U - V_) \] --- ## Step 4: Express Mass Change The mass of the rocket **decreases**: \[ \frac{dM}{dt} = -\dot{m} \] --- ## Step 5: Substitute and Simplify Plug in \( \frac{dM}{dt} = -\dot{m} \): \[ M \frac{dU}{dt} + U (-\dot{m}) = -D - \dot{m}(U - V_) \] \[ M \frac{dU}{dt} - U \dot{m} = -D - \dot{m} U + \dot{m} V_ \] \[ M \frac{dU}{dt} - U \dot{m} + U \dot{m} = -D + \dot{m} V_ \] \[ M \frac{dU}{dt} = -D + \dot{m} V_ \] --- ## Step 6: Solve for Acceleration \( a \) \[ M a = \dot{m} V_ - D \] \[ \boxed{ a = \frac{\dot{m} V_ - D}{M} } \] --- ## Step 7: **Explanation of Terms** - \( \dot{m} V_ \): **Thrust** (momentum per time from exhaust) - \( D \): **Drag** (air resistance) - \( M \): **Instantaneous mass** of rocket --- ## **Final Answer** \[ \boxed{ a = \frac{\dot{m} V_ - D}{M} } \] - **Control volume:** Rocket itself (moving CV, mass is leaving through exhaust) - **Why:** The momentum flux due to exhaust and the external drag are the only forces acting horizontally. - **Acceleration** is the net force (thrust minus drag) divided by the current rocket mass.