Concept: According to Newton’s second law of motion, the net force acting on a body is equal to the product of its mass and acceleration: \[ F = m a \]
To find the acceleration, we use the definition of acceleration as the rate of change of velocity: \[ a = \frac{v - u}{t} \]
where \(v\) is the final velocity, \(u\) is the initial velocity, and \(t\) is the time. Here, the car comes to rest, so \(v = 0\), \(u = 20~m/s\), \(t = 10~s\).


Calculation: \[ a = \frac{0 - 20}{10} = -2~m/s^2 \]
The negative sign indicates that the acceleration is opposite to the direction of motion (deceleration). Using \(F = m a\): \[ F = 1000 \times (-2) = -2000~N \]
The magnitude of the force is \(2000~N\). This is the average braking force needed to stop the car over 10 seconds.


Explanation: The car decelerates uniformly under the action of the brakes. The negative sign shows the force acts opposite to motion. Using Newton's law ensures the correct force magnitude is obtained regardless of the vehicle’s mass.
Quick Tip: Always calculate acceleration first, considering the direction of motion. Then apply \(F = m a\). The magnitude of the force is what matters when reporting results. Keep units consistent throughout the calculation.

Concept: Work done against gravity is the product of the weight of the person and the vertical height climbed: \[ W = m g h \]
where \(m\) is mass, \(g\) is gravitational acceleration (\(9.8~m/s²\)), and \(h\) is height. This formula comes from the general definition of work \(W = F \cdot d \cdot \cos \theta\), where force \(F\) is along the displacement (\(\theta = 0°\) for vertical climb).


Calculation: \[ W = 50 \times 9.8 \times 10 = 4900~J \approx 5000~J \]

Explanation: The person’s weight acts downward, and the person moves upward against gravity. Therefore, the work done is stored as gravitational potential energy. The calculation assumes no energy losses due to friction or air resistance.
Quick Tip: Work against gravity is equivalent to the gain in potential energy. Multiply the weight of the person by the vertical height climbed to find the work done. Ensure consistent units for mass and height.

Concept: An object moving in a circular path experiences a centripetal force directed toward the center. The magnitude of this force is: \[ F_c = \frac{m v^2}{r} \]
where \(m\) is mass, \(v\) is speed, and \(r\) is radius of the circle. This formula comes from equating centripetal acceleration \(a_c = v^2/r\) to \(F = m a\).


Calculation: \[ F_c = \frac{800 \times 20^2}{50} = \frac{800 \times 400}{50} = 6400~N \]

Explanation: The car must constantly change direction to follow the circular path. The required inward force is supplied by friction between tires and road. If the centripetal force exceeds friction, the car skids outward.
Quick Tip: Centripetal force depends on mass, speed squared, and radius. It always acts toward the center of the circle. Ensure correct units and use this formula only for uniform circular motion.

Concept: For incompressible fluids, the volume flow rate is constant along the pipe: \[ A_1 v_1 = A_2 v_2 \]
where \(A = \pi r^2\) is the cross-sectional area. This is the continuity equation.


Calculation: \[ A_1 = \pi (0.02)^2 = 1.256 \times 10^{-3}~m², \quad A_2 = \pi (0.01)^2 = 3.14 \times 10^{-4}~m² \] \[ v_2 = \frac{A_1}{A_2} v_1 = \frac{1.256 \times 10^{-3}}{3.14 \times 10^{-4}} \times 5 \approx 20~m/s \]

Explanation: As the pipe narrows, the velocity must increase to maintain constant flow. This principle applies to incompressible, non-viscous fluids.
Quick Tip: Use the continuity equation \(A v = constant\). A smaller cross-section means higher velocity. Always convert radii to meters for consistency.

Concept: The heat lost by hot water equals heat gained by cold water (conservation of energy): \[ m_1 c (T_1 - T_f) = m_2 c (T_f - T_2) \]
Here, \(c\) is specific heat of water, which cancels out.


Calculation: \[ 200 (80 - T_f) = 100 (T_f - 20) \] \[ 16000 - 200 T_f = 100 T_f - 2000 \] \[ 18000 = 300 T_f \implies T_f = 60~°C \]

Explanation: The final temperature is closer to the hot water temperature because the mass of hot water is greater. This is an example of mixing with no heat loss.
Quick Tip: Heat lost by hot water equals heat gained by cold water. The larger mass influences the final temperature. Ensure that the final temperature is between initial temperatures.

Concept: According to Ohm's law, \[ V = I R \implies I = \frac{V}{R} \]
where \(V\) is voltage, \(R\) is resistance, and \(I\) is current.


Calculation: \[ I = \frac{10}{5} = 2~A \]

Explanation: The resistor allows 2 A of current to flow. Ohm’s law shows the direct relationship between voltage, current, and resistance. This assumes ideal resistor with no temperature change.
Quick Tip: Use Ohm’s law \(I = V/R\) with consistent units. Current flows from positive to negative terminal. Always verify voltage and resistance values before calculating.

Concept: pH is calculated using the formula \( pH = -\log_{10}[H^+]\). HCl is a strong acid, so it dissociates completely, and \([H^+] = [HCl]\).


Calculation: \[ pH = -\log_{10}(0.001) = -\log_{10}(10^{-3}) = 3 \]

Explanation: The pH value represents the concentration of hydrogen ions. A 0.001 M HCl solution is acidic, and its pH is 3.
Quick Tip: For strong acids and bases, assume complete dissociation. Use \( pH = -\log[H^+] \). Double-check units and use base-10 logarithms for pH calculations.

Concept: From the balanced equation, 2 moles of Na produce 2 moles of NaCl, so mole ratio is 1:1. Use \( mass = moles \times molar mass \). Molar mass of NaCl = 58.5 g/mol.


Calculation: \[ Mass of NaCl = 2 \times 58.5 = 117.0~g \]

Explanation: All 2 moles of sodium react completely with chlorine. Using molar mass and mole ratio, the total mass of NaCl formed is 117.0 g.
Quick Tip: Always check the mole ratio from the balanced chemical equation. Multiply moles of product by molar mass to find the mass formed. Ensure units are consistent.

Concept: Use Boyle’s Law for constant temperature: \(P_1 V_1 = P_2 V_2\). Pressure and volume are inversely proportional.


Calculation: \[ V_2 = \frac{P_1 V_1}{P_2} = \frac{1.0 \times 5.0}{2.0} = 2.5~L \]

Explanation: Doubling the pressure halves the volume. This demonstrates the inverse relationship between pressure and volume at constant temperature.
Quick Tip: Apply Boyle’s Law \(P_1 V_1 = P_2 V_2\) for isothermal conditions. Solve for unknown volume or pressure, keeping units consistent. This relationship is inversely proportional.

Concept: Cell potential \(E_cell = E_cathode - E_anode\). Here, Ag is oxidized (anode) and Cu is reduced (cathode). Rearranging: \(E_Cu = E_cell + E_Ag = 0.46 - 0.80 = -0.34~V\). But standard reduction potential is positive for reduction: \(E_Cu = +0.34 V\).


Explanation: Using the relation between cell potential and electrode potentials, the standard reduction potential of Cu is determined to be 0.34 V.
Quick Tip: Identify cathode and anode correctly. Use \(E_cell = E_cathode - E_anode\). Sign conventions are crucial; check whether reduction or oxidation is considered.

Concept: Rate law: Rate = k[A]\(^2\). Changing [A] by factor of 3, new rate = k(3[A])\(^2\) = 9 k[A]\(^2\) = 9 × original rate.


Explanation: The reaction is second order with respect to A, so the rate depends on the square of the concentration. Tripling [A] increases rate 9-fold.
Quick Tip: Always apply exponents in the rate law when concentration changes. Multiply the effects for each reactant to determine overall rate change.

Concept: Oxygen in photosynthesis is produced when water molecules are split (photolysis) by light energy in photosystem II. Calvin cycle fixes CO\(_2\) but does not release O\(_2\).


Explanation: Light energy splits H\(_2\)O into protons, electrons, and O\(_2\). Oxygen is released as a by-product during the light-dependent reaction.
Quick Tip: Remember: O\(_2\) comes from water photolysis, not CO\(_2\) fixation. Calvin cycle produces glucose, while oxygen is generated in light reactions.

Concept: This is a monohybrid cross. Use a Punnett square to determine offspring genotypes. Genotypes: TT, Tt, tT, tt. Only tt results in short plants.


Calculation: \[ Probability of short (tt) = \frac{1}{4} \]

Explanation: Out of 4 possible genotypes, one (tt) is short. Hence, the probability is 1/4.
Quick Tip: Draw a Punnett square for clarity. Dominant allele masks recessive in heterozygotes. Probability = favorable outcomes / total outcomes.

Concept: The cerebellum coordinates voluntary movements, posture, and balance. Cerebrum handles conscious thought, medulla controls involuntary actions, and thalamus acts as a relay.


Explanation: The cerebellum integrates sensory input and motor commands to maintain smooth, coordinated movement and equilibrium.
Quick Tip: Remember: Cerebellum = balance & coordination. Cerebrum = thinking, Medulla = involuntary functions, Thalamus = sensory relay. Focus on function rather than size.

Concept: Transpiration is the loss of water vapor from plant leaves. It maintains water flow, nutrient transport, and helps in temperature regulation.


Explanation: Evaporation of water from stomata cools the plant and drives the ascent of sap. It does not occur only in roots; roots absorb water, leaves transpire.
Quick Tip: Transpiration primarily occurs through leaves, not roots. It cools the plant, maintains water and nutrient flow. Always distinguish between absorption and loss.

Concept: Estrogen is the primary female sex hormone responsible for secondary sexual characteristics, such as breast development, body fat distribution, and menstrual cycle regulation. Progesterone aids in reproductive cycle but is not responsible for secondary sexual traits.


Explanation: Estrogen triggers physical changes at puberty in females, differentiating from male characteristics and preparing the body for reproduction.
Quick Tip: Remember: Estrogen = female secondary sexual characteristics. Testosterone = male characteristics. Progesterone regulates pregnancy and menstrual cycle.