Chapter 3 Solved Numericals: Dynamics

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Chapter Overview & SLOs

What is covered in these solved numericals? This section provides detailed step-by-step solutions to the numerical problems for Chapter 3, "Dynamics." The exercises focus on applying Newton's laws and the principles of momentum to real-world scenarios. How do we calculate net force and acceleration? You will learn to calculate total force required to move objects against gravity and determine acceleration using Newton's Second Law (F = ma): - Force (F) measured in Newtons (N = kg·m/s²) - Mass (m) in kilograms (kg) - remember to convert grams to kg (divide by 1000) - Acceleration (a) in m/s² - For vertical motion: Force = mass × (g ± a) depending on direction of motion How do we calculate linear momentum and rate of change of momentum? You will learn to calculate: - Linear momentum: $p = mv$ (momentum = mass × velocity). Units: kg·m/s or Ns (Newton-second) - Rate of change of momentum: $F = \frac{\Delta p}{\Delta t} = \frac{mv_f - mv_i}{t}$ (force equals change in momentum divided by time). This is an alternative form of Newton's Second Law. - Finding the force applied by obstacles (like sand or cushions) to stop moving objects - Understanding that a negative force value indicates a direction opposite to the motion How do we apply conservation of momentum to gun recoil? A significant portion of these numericals covers the Law of Conservation of Momentum, specifically calculating the recoil velocity of a gun after firing a bullet: - Before firing: Total momentum = 0 (gun and bullet at rest) - After firing: $m_b v_b + m_g v_g = 0$ - Recoil velocity formula: $v_g = -\frac{m_b v_b}{m_g}$ (negative sign indicates opposite direction) - Unit conversions: Remember to convert grams to kilograms for bullet mass (divide by 1000) How do we determine stopping time from braking force? You will learn to calculate the time required to stop a moving object when a constant opposing (braking) force is applied: - Using $F = \frac{m(v_f - v_i)}{t}$ where $v_f = 0$ (final velocity when stopped) - Rearranging: $t = \frac{mv_i}{F}$ (since $v_f = 0$) - Understanding that braking force is negative relative to direction of motion These solutions emphasize correct unit usage (Newtons and Newton-seconds) and the physical interpretation of negative signs in opposing forces. They are strictly designed to help students master the mathematical requirements of the FBISE 2026 annual examination.

• How do we calculate net force and acceleration? Calculate net force and acceleration by applying Newton's Second Law (F = ma) in both horizontal planes (pushing/pulling objects) and vertical planes (lifting objects against gravity), remembering to convert mass to kilograms (divide grams by 1000).
• How do we calculate linear momentum and rate of change of momentum? Solve problems to determine the linear momentum of a body using p = mv (units: kg·m/s or Ns) and relate force to the rate of change of momentum using F = Δp/Δt = (mv_f - mv_i)/t, understanding that negative force indicates direction opposite to motion.
• How do we apply conservation of momentum to gun recoil? Apply the Law of Conservation of Momentum (total momentum before = total momentum after) to calculate the recoil velocity of firearms using v_g = -m_b v_b / m_g, and determine final velocities in isolated systems such as collisions and explosions, remembering to convert bullet mass from grams to kilograms.
• How do we determine stopping time from braking force? Determine the time required to stop a moving object when a constant opposing (braking) force is applied using t = mv_i / F (derived from F = m(v_f - v_i)/t with v_f = 0), and understand the relationship between braking force, initial velocity, mass, and stopping time.

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