Physics 2021 Jamb Examination

In a displacement-time graph, the gradient or slope at any point represents the rate of change of displacement with respect to time. This rate of change is defined as velocity. When the graph is a straight line, the gradient is constant throughout the motion, which implies that the velocity does not change over time. Consequently, a straight-line displacement-time graph represents uniform velocity. A horizontal line would indicate zero velocity (rest), while a curved line would indicate changing velocity (acceleration).

Pressure is defined as the force acting per unit area. In this scenario, the force exerted by the man is his weight, which remains constant regardless of his posture. The relationship for pressure is expressed as: \[ P = \frac{F}{A} \] Since the force \( F \) is constant, pressure \( P \) is inversely proportional to the area of contact \( A \). To achieve the greatest pressure, the area of contact must be as small as possible. Lying flat or on the belly involves a large surface area, leading to low pressure. Standing on both feet reduces the area, but standing on the toes of only one foot provides the minimum possible contact area, thereby exerting the maximum pressure on the ground.

Fundamental quantities are independent physical quantities that are not defined in terms of other quantities (e.g., mass, length, time). Derived quantities, however, are created through the mathematical combination of fundamental quantities.
1. Area is derived from length multiplied by length (\( \text{m}^2 \)).
2. Thrust is a type of force, which is mass multiplied by acceleration (\( \text{kg} \cdot \text{m/s}^2 \) or Newtons).
3. Pressure is derived from force divided by area (\( \text{N/m}^2 \) or Pascals).
4. Mass is one of the basic fundamental quantities in the SI system and its unit (kg) is not derived from others. Since Area, Thrust, and Pressure are all built from fundamental units, only I, II, and III are derived quantities.

This is a case of a perfectly inelastic collision where the two masses coalesce and move with a common velocity after impact. According to the principle of conservation of linear momentum, the total momentum before collision equals the total momentum after collision.
Let:
\( m_1 = 0.5 \, \text{kg} \)
\( u_1 = 10 \, \text{ms}^{-1} \)
\( m_2 = 0.5 \, \text{kg} \)
\( u_2 = 0 \, \text{ms}^{-1} \) (at rest)
\( v \) = common velocity \[ m_1u_1 + m_2u_2 = (m_1 + m_2)v \] Substituting the values: \[ (0.5 \times 10) + (0.5 \times 0) = (0.5 + 0.5)v \] \[ 5 + 0 = 1.0v \] \[ v = 5.0 \, \text{ms}^{-1} \] The momentum of the first ball is shared between the combined masses, resulting in a common velocity of 5.0ms\(^{-1}\).

Simple Harmonic Motion (SHM) is a specific type of periodic motion defined by two essential conditions. First, the acceleration of the body must be directly proportional to its displacement from a fixed central position (equilibrium point). Second, the direction of this acceleration must always be opposite to the direction of displacement, meaning it is always directed toward the fixed equilibrium point. The mathematical definition is: \[ a = -\omega^2x \] While the motion often occurs in a straight line, that is not the defining characteristic. The proportionality to the displacement (not the square of the distance) and the constant direction towards the fixed point are what define the motion as simple harmonic.

The kinetic theory of gases describes the behaviour of gas molecules based on several postulates. These include the ideas that gas molecules are in constant, random motion (A), they possess a distribution of speeds (B), and the absolute temperature of the gas is directly proportional to the average kinetic energy of the molecules (C). However, one of the key assumptions of an ideal gas is that the collisions between the molecules, and between the molecules and the container walls, are perfectly elastic. This means that there is no net loss of total kinetic energy during the collisions. The statement that collisions are perfectly inelastic is incorrect, as inelastic collisions would result in a loss of kinetic energy and the gas would eventually lose its thermal energy.

Since the volume of the gas remains constant, we apply the Pressure Law (Gay-Lussac's Law), which states that the pressure of a fixed mass of gas is directly proportional to its absolute temperature. The temperatures must be converted from Celsius to Kelvin. \[ T_1 = 47 + 273 = 320 \, \text{K} \] \[ T_2 = 27 + 273 = 300 \, \text{K} \] \[ P_1 = 80 \, \text{Nm}^{-2} \] Using the formula: \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \] \[ \frac{80}{320} = \frac{P_2}{300} \] \[ \frac{1}{4} = \frac{P_2}{300} \] \[ P_2 = \frac{300}{4} = 75 \, \text{Nm}^{-2} \] The reduction in temperature leads to a decrease in the kinetic energy of the molecules, resulting in fewer and less forceful collisions, thereby reducing the pressure to 75 Nm\(^{-2}\).

The total heat extracted (\( Q_T \)) consists of two parts: the heat removed to cool the water from 10ºC to 0ºC (\( Q_1 \)), and the latent heat removed to freeze the water at 0ºC (\( Q_2 \)).
\( Q_T = 88000 \, \text{J} \)
\( m = 0.5 \, \text{kg} \)
\( c_w = 4200 \, \text{Jkg}^{-1}\text{C}^{-1} \)
\( \Delta\theta = 10 - 0 = 10 \, ^\circ\text{C} \) \[ Q_1 = mc_w\Delta\theta \] \[ Q_1 = 0.5 \times 4200 \times 10 = 21000 \, \text{J} \] Now, find the heat used for the phase change (\( Q_2 \)): \[ Q_T = Q_1 + Q_2 \] \[ 88000 = 21000 + Q_2 \] \[ Q_2 = 88000 - 21000 = 67000 \, \text{J} \] Since \( Q_2 = mL_f \): \[ 67000 = 0.5 \times L_f \] \[ L_f = \frac{67000}{0.5} = 134000 \, \text{Jkg}^{-1} \] \[ L_f = 134.0 \, \text{kJkg}^{-1} \] The specific latent heat of fusion of ice is therefore 134 kJ/kg.

Relative humidity is the measure of the amount of water vapour present in the air compared to the maximum amount the air could hold at that temperature. The instrument specifically designed to measure this is the hygrometer. A common type is the wet-and-dry bulb hygrometer. Other instruments listed serve different purposes: a hydrometer measures the relative density of liquids, a manometer measures gas pressure, and a hypsometer is used to determine the boiling point of liquids or to measure altitude based on boiling point.

The relationship between the speed of a wave (\( v \)), its frequency (\( f \)), and its wavelength (\( \lambda \)) is given by the wave equation: \[ v = f\lambda \] The period (\( T \)) is the reciprocal of the frequency: \[ T = \frac{1}{f} \] Combining these equations, we get: \[ v = \frac{\lambda}{T} \quad \Rightarrow \quad T = \frac{\lambda}{v} \] Given: \( \lambda = 1.65 \, \text{m} \) \( v = 330 \, \text{ms}^{-1} \) \[ T = \frac{1.65}{330} \] \[ T = \frac{1.65}{1.65 \times 200} = \frac{1}{200} \] \[ T = 0.005 \, \text{s} \] The time taken for one complete vibration of the sound wave in air is 0.005 seconds.