1. **Problem 1: Measurement and Dimensions** State the problem: Given a physical quantity, find its dimensional formula. Formula: Use the fundamental dimensions: Mass ($M$), Length ($L$), Time ($T$). Example: Velocity has dimensions of length per time, so $[V] = LT^{-1}$. Explanation: Dimensional analysis helps verify equations and convert units. 2. **Problem 2: Friction** State the problem: Calculate the force of friction acting on a body. Formula: $F_f = \mu N$, where $\mu$ is the coefficient of friction and $N$ is the normal force. Explanation: Friction opposes motion; static friction prevents motion, kinetic friction acts during motion. 3. **Problem 3: Linear Momentum Conservation** State the problem: Analyze a collision where two bodies interact. Formula: Total momentum before collision equals total momentum after collision: $$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$$ Explanation: Momentum is conserved in isolated systems without external forces. 4. **Problem 4: Boat and Person Problem** State the problem: A person walks on a boat; find the boat's displacement. Formula: Using conservation of momentum and center of mass: $$m_p x_p + m_b x_b = 0$$ Explanation: The boat moves opposite to the person to conserve momentum. 5. **Problem 5: Mechanical Energy Conservation** State the problem: Calculate velocity of a body falling under gravity without friction. Formula: Total mechanical energy conserved: $$PE_i + KE_i = PE_f + KE_f$$ Example: $$mgh = \frac{1}{2}mv^2$$ Explanation: Potential energy converts to kinetic energy. **Bonus Questions:** 6. **Bonus 1: Dimension Equation Building** State the problem: Derive the dimensional formula for force. Solution: Force = mass × acceleration $$[F] = M \times LT^{-2} = MLT^{-2}$$ 7. **Bonus 2: Friction and Inclined Plane** State the problem: Find minimum angle for an object to start sliding on an inclined plane with friction coefficient $\mu$. Formula: $$\tan \theta = \mu$$ Explanation: At this angle, component of gravity overcomes friction. 8. **Bonus 3: Elastic Collision in One Dimension** State the problem: Find final velocities after elastic collision. Formula: $$v_1' = \frac{(m_1 - m_2)v_1 + 2m_2 v_2}{m_1 + m_2}$$ $$v_2' = \frac{(m_2 - m_1)v_2 + 2m_1 v_1}{m_1 + m_2}$$ Explanation: Both momentum and kinetic energy conserved.