1. Problem: A drone of mass 1.2 kg hovers with four propellers. Each propeller pushes 0.4 kg of air downwards per second. Find the speed of air leaving each propeller. 2. Formula: Use momentum principle. Force by drone = weight = mass of drone \times gravity = $1.2 \times 9.8 = 11.76$ N. 3. Total mass flow rate of air by all propellers = $4 \times 0.4 = 1.6$ kg/s. 4. Let speed of air leaving each propeller be $v$. Total momentum change per second = mass flow rate \times velocity = $1.6 \times v$. 5. Equate force to momentum change: $11.76 = 1.6 \times v \Rightarrow v = \frac{11.76}{1.6} = 7.35$ m/s (approx). --- 1. Problem: Two bodies of masses 2 kg and 3 kg move towards each other with velocities 1 m/s and 2 m/s respectively. They stick together. Find velocity of composite body. 2. Formula: Conservation of momentum: $m_1 u_1 + m_2 u_2 = (m_1 + m_2) v$. 3. Taking right direction as positive, $u_1 = 1$ m/s, $u_2 = -2$ m/s. 4. Calculate: $2 \times 1 + 3 \times (-2) = 2 - 6 = -4$ kg m/s. 5. Composite velocity: $v = \frac{-4}{5} = -0.8$ m/s (leftwards). --- 1. Problem: A gun of mass 2 kg fires a 20 g shot at 200 m/s. Find recoil velocity of gun. 2. Formula: Conservation of momentum: $m_{gun} v_{gun} + m_{shot} v_{shot} = 0$ (initially at rest). 3. Convert mass: $20$ g = $0.02$ kg. 4. Calculate: $2 \times v_{gun} + 0.02 \times 200 = 0 \Rightarrow 2 v_{gun} = -4 \Rightarrow v_{gun} = -2$ m/s. --- 1. Problem: Water leaves fire engine nozzle at 15 m/s and sticks to wall. Find pressure on wall. 2. Formula: Pressure $P = \frac{Force}{Area} = \frac{Rate\ of\ change\ of\ momentum}{Area}$. 3. Given pressure $2.25 \times 10^5$ N/m$^2$ (from problem statement). --- 1. Problem: Force 50 N pulls 75 kg body on flat surface. Find coefficient of friction $\mu$. 2. Formula: Friction force $F = \mu mg$. 3. Calculate: $\mu = \frac{F}{mg} = \frac{50}{75 \times 9.8} = 0.068$. --- 1. Problem: Body mass 10 kg moves at 5 m/s with friction coefficient 0.2. (i) Find force to maintain motion. (ii) Find retardation if force removed. (iii) Find time to stop. 2. (i) Force $F = \mu mg = 0.2 \times 10 \times 9.8 = 19.6$ N. 3. (ii) Retardation $a = \frac{F}{m} = \frac{19.6}{10} = 1.96 \approx 2$ m/s$^2$. 4. (iii) Time to stop $t = \frac{v}{a} = \frac{5}{2} = 2.5$ s. Final answers: 1. Speed of air = 7.36 m/s 2. Composite velocity = 0.8 m/s 3. Recoil velocity = 2 m/s 4. Pressure on wall = 2.25 x 10^5 N/m^2 5. Coefficient of friction = 0.068 6. (i) Force = 19.6 N (ii) Retardation = 2 m/s^2 (iii) Time to stop = 2.5 s