Kinetics

Step 1: The root mean square speed ( ) of particles in Brownian motion is given by the equation: where: - is the Boltzmann constant, - is the temperature in Kelvin, - is the mass of the smoke particle. Step 2: Since the temperature at NTP (Normal Temperature and Pressure) is , we can substitute these values into the equation. Step 3: Simplify the expression: Converting to millimeters per second: Thus, the root mean square speed is approximately .

Step 1: The impedance in a series LCR circuit is given by:
where and .
Step 2: At resonance frequency , the inductive and capacitive reactances cancel:

Step 3: For frequencies below and above , follows a characteristic curve, which is represented by option (C).

Step 1: Using Einstein's photoelectric equation:
where .
Step 2: Given , we get:

Step 3: Substituting in the equation:

Step 4: Since , we conclude:

The problem involves calculating the time taken for an object to fall from a certain height under gravity. Here, we use the equation of motion for free fall without initial velocity:
h = ¹⁄₂ gt²

Where:

• h = height (45 m)
• g = acceleration due to gravity (9.8 m/s²)
• t = time in seconds

Substitute the values into the equation:
45 m = ¹⁄₂ (9.8 m/s²) t²
Solving for t²:
45 m = 4.9t²
t² = ^{45 m}⁄_{4.9 m/s²}
t² = 9.18
t ≈ √9.18 ≈ 3.03 s

Rounding to the nearest second, t ≈ 3 s.

Therefore, the time taken for the stone to reach the ground is approximately 3 seconds, confirming the correct answer is 3 s.

Step 1: Use the formula for potential energy The potential energy ( ) of a body at a height is given by the formula: where: - is the potential energy, - is the mass of the body, - is the acceleration due to gravity, - is the height. Step 2: Substitute the given values Given: - , - , - . Now substitute into the formula: Answer: Therefore, the mass of the body is 2 kg. So, the correct answer is option (1).

Step 1: Use the formula for kinetic energy The formula for the kinetic energy ( ) of a body is: where: - is the mass of the body, - is the velocity of the body. Step 2: Substitute the given values Given: - Mass , - Velocity . Substitute these values into the formula: Answer: Therefore, the kinetic energy of the body is 9 J. So, the correct answer is option (2).